answer：Okay, I need to write a Python function to format a book title according to specific rules. Let's see what the requirements are.First, the function should take a single string as input. The output should be a formatted string based on four rules.Rule 1: The first letter of each word should be capitalized. So, each word's first letter is uppercase, and the rest are lowercase. For example, "the" becomes "The".Rule 2: All other letters should be lowercase. So, any letters after the first in a word should be lowercase. Like "GATSBY" becomes "Gatsby".Rule 3: The entire title should be enclosed in square brackets. So, the formatted string starts with [ and ends with ].Rule 4: Leading or trailing spaces should be removed. So, I need to strip whitespace from both ends of the input string.Let me think about how to approach this step by step.First, I'll need to process the input string. I should start by stripping any leading or trailing spaces. That's easy with the strip() method.Next, I need to split the string into words. Using the split() method without arguments will split on any whitespace, which handles multiple spaces between words.Then, for each word in the list, I need to capitalize the first letter and make the rest lowercase. Wait, the capitalize() method in Python does exactly that. So, for each word, I can apply word.capitalize().After processing each word, I'll join them back into a single string with spaces between them.Once I have the properly capitalized title, I need to enclose it in square brackets. So, I'll add '[' at the beginning and ']' at the end.Putting it all together:1. Strip the input string.2. Split into words.3. Capitalize each word.4. Join the words with spaces.5. Enclose in square brackets.Let me test this logic with the example given. The input is " the great gatsby ". After stripping, it becomes "the great gatsby". Splitting into words gives ["the", "great", "gatsby"]. Capitalizing each gives ["The", "Great", "Gatsby"]. Joined with spaces becomes "The Great Gatsby". Enclosing in brackets gives "[The Great Gatsby]", which matches the expected output.What about other cases? For example, if the input is all uppercase, like " PYTHON PROGRAMMING ". After stripping, it's "PYTHON PROGRAMMING". Splitting gives ["PYTHON", "PROGRAMMING"]. Capitalizing each gives ["Python", "Programming"]. Joined as "Python Programming", enclosed in brackets.Another test case: input is " hello world ". After stripping, "hello world". Split into ["hello", "world"], capitalize each to ["Hello", "World"], joined as "Hello World", enclosed as "[Hello World]".What if the input is an empty string? Well, stripping would leave an empty string, but the function should return "[]" in that case? Or maybe the function expects a non-empty string. The problem statement says it's a book title, so perhaps it's assumed to be non-empty, but I should handle that case gracefully.Wait, if the input is an empty string after stripping, then the function would return "[]". That seems correct.Another edge case: a single word. For example, input " alice ". After stripping, "alice". Capitalize to "Alice". Enclosed as "[Alice]".What about words with apostrophes or hyphens? Like "don't look now" or "brother-in-law". The capitalize method should handle them correctly because it only affects the first character and the rest are lowercased. So, "don't" becomes "Don't", "brother-in-law" becomes "Brother-In-Law"? Wait, no. Wait, the capitalize method makes the first character uppercase and the rest lowercase. So, "brother-in-law" would become "Brother-in-law", which is correct because the hyphen doesn't affect the capitalization of the following letters. So, the function should handle such cases correctly.So, the steps in code would be:- Strip the input string.- Split into words.- Capitalize each word.- Join with spaces.- Enclose in square brackets.Now, implementing this in Python.Function steps:def format_book_title(title): # Step 1: Strip leading/trailing spaces stripped = title.strip() # Step 2: Split into words words = stripped.split() # Step 3: Capitalize each word capitalized = [word.capitalize() for word in words] # Step 4: Join into a string formatted_title = ' '.join(capitalized) # Step 5: Enclose in brackets return f'[{formatted_title}]'Wait, but what if the stripped string is empty? For example, if the input is all spaces. Then, words would be an empty list, and ' '.join would be an empty string. So the function would return "[]", which is correct.Testing this function with the example:Input: " the great gatsby "After stripping: "the great gatsby"Split into ["the", "great", "gatsby"]Capitalized: ["The", "Great", "Gatsby"]Joined: "The Great Gatsby"Enclosed: "[The Great Gatsby]" → correct.Another test case: input is " hello world " → output "[Hello World]".Another test case: input is " " → output "[ ]"? Wait no, because stripped would be an empty string, words would be an empty list, so ' '.join would be empty string. So the function returns "[]", which is correct.Wait, no. If stripped is an empty string, then words = stripped.split() → words is an empty list. So capitalized is an empty list. ' '.join(capitalized) is ''. So the function returns '[]', which is correct.Another test case: input is " a " → output "[A]".Yes, that should work.So the function seems to handle all cases correctly.

answer：Okay, I need to write a Python function that checks if any of the book titles in a given list contain certain keywords related to antique books. The keywords are "antique", "vintage", "classic", "historical", "retro", "old", "ancient". The function should return a list of titles that have at least one of these keywords, and it should be case-insensitive.Hmm, so first, I should think about how to approach this. Let's see. I'll start by defining the function, maybe call it find_antique_books, which takes a list of strings as input.Next, I need to process each title in the list. For each title, I should check if any of the keywords are present. But since the check is case-insensitive, I should probably convert both the title and the keywords to lowercase before comparing.Wait, but the keywords are fixed, so maybe I can create a set of lowercase keywords. That way, for each title, I can convert it to lowercase and then check if any of the keywords are in it.So the steps are:1. Define the list of keywords in lowercase.2. Iterate over each title in the input list.3. For each title, convert it to lowercase.4. Check if any keyword is present in the lowercase title.5. If yes, add the original title (not the lowercase one) to the result list.6. Return the result list.Let me think about how to implement this in Python. So, the keywords can be a set for faster lookups, but in this case, since each title is checked against all keywords, perhaps using a set isn't necessary. Alternatively, for each title, I can loop through each keyword and see if it's present.Wait, but for each title, I can create a lowercase version and then check if any of the lowercase keywords are in it. So, perhaps for each title, I can do something like:lower_title = title.lower()if any(keyword in lower_title for keyword in keywords): add to result.Yes, that makes sense. So the keywords are in a list, and for each title, we check if any keyword is a substring of the lowercase title.So, putting it all together:Define the function, create the keywords list as lowercase, then loop through each title, check for any keyword presence, and collect the matching titles.Let me test this logic with the example given.The example input is:book_titles = [ "Modern Python Programming", "Classic Tales of Adventure", "Historical Narratives from the Past", "The Art of Vintage Photography", "Ancient Civilizations", "Contemporary Art"]The keywords are "antique", "vintage", "classic", "historical", "retro", "old", "ancient".So, the function should return the four titles that contain these keywords.Testing each title:1. "Modern Python Programming" - no keywords.2. "Classic" is in the title, so it's included.3. "Historical" is present.4. "Vintage" is present.5. "Ancient" is present.6. "Contemporary" doesn't match any keyword.So the output is as expected.Now, what about case insensitivity? For example, if a title is "Classic", "CLASSIC", or "Classic", it should still match. Since we're converting the title to lowercase and the keywords are lowercase, it will correctly match regardless of the case in the title.Another test case: a title with a keyword in uppercase, like "ANTIQUE Furniture". The function should include it.What about partial words? Like "antiquing" or "vintager"? Wait, the problem says "contains any of the specified keywords". So it's a substring match. So if a title has "antiquing", it contains "antique" as a substring, right? Wait, no. Because "antiquing" has 'antiqu' but not 'antique'. Wait, no, 'antiquing' is 'antique' plus 'ing'? Wait, no. Wait, 'antique' is 7 letters, 'antiquing' is 9 letters. So 'antique' is a substring of 'antiquing'? Let's see: 'antiquing' starts with 'antiqu' (6 letters), then 'i', 'n', 'g'. So 'antique' is 7 letters, so 'antiquing' does not contain 'antique' as a substring. So the function would not match 'antiquing' for the keyword 'antique'.Wait, but if the title is "Antiquing", then the lowercase is "antiquing". The keyword is "antique". So 'antiquing' does not contain 'antique' as a substring. So it won't match.So the function is checking for exact keyword substrings, case-insensitive.So that's correct.Now, how to implement this in code.First, the function:def find_antique_books(titles): keywords = ["antique", "vintage", "classic", "historical", "retro", "old", "ancient"] result = [] for title in titles: lower_title = title.lower() for keyword in keywords: if keyword in lower_title: result.append(title) break # no need to check other keywords once one is found return resultWait, but this would loop through each keyword for each title until a match is found. Alternatively, using any() with a generator expression would be more efficient.So perhaps:def find_antique_books(titles): keywords = {"antique", "vintage", "classic", "historical", "retro", "old", "ancient"} result = [] for title in titles: lower_title = title.lower() if any(keyword in lower_title for keyword in keywords): result.append(title) return resultWait, but using a set for keywords might be more efficient, but in this case, it's not a big deal since the keywords are only 7.Alternatively, using a set for the keywords and then checking if any keyword is a substring.Wait, but in the code above, the any() function will return True as soon as one keyword is found in the title, so it's efficient.So this should work.Testing the example:The function should return the four titles as expected.Another test case: a title with multiple keywords, like "Old Vintage Classic". It should be included.What about a title that doesn't have any keywords, like "New Book"? It should be excluded.What about a title that has a keyword in a different case, like "Classic"? It should be included.Yes.So the function should handle all these cases.I think this should work.

answer：Okay, I need to write a Python function that finds the most frequently occurring word in a given string. Hmm, let's think about how to approach this.First, the input is a string that can have punctuation and mixed cases. So, I need to process this string to extract words, ignoring case and punctuation. Oh right, the output should be in lowercase.So the steps I think are:1. Normalize the string: convert everything to lowercase.2. Remove punctuation. But wait, how? Maybe replace all non-alphanumeric characters with spaces. Or split on word boundaries.3. Split the string into words. But what defines a word here? Probably, words are separated by whitespace, and any sequence of letters is a word, ignoring punctuation attached to them.Wait, maybe I can use regular expressions to find all the words. Like, using re.findall to get all sequences of letters, ignoring case.Yes, that makes sense. So, I can use a regex pattern that matches word characters, but I need to consider apostrophes? Or not? The problem statement doesn't specify, but in the example, words like "don't" might be treated as one word. But the example given doesn't have such cases. Hmm, but the problem says to ignore punctuation, so perhaps any punctuation attached to words should be stripped.Alternatively, perhaps the regex should match any sequence of letters, regardless of what's around them. So, perhaps the pattern [a-zA-Z]+ would capture all words, ignoring case.Wait, but the example given: "The quick brown fox jumps over the lazy dog. The dog barks loudly." The words are "The", "quick", etc. So when we process it, "The" becomes "the", and appears twice.So the plan is:- Convert the entire string to lowercase.- Use regex to find all sequences of letters, ignoring any other characters.- Count the occurrences of each word.- Find the word with the highest count. If there's a tie, return any.So, step by step:1. Import necessary modules. I think I'll need re for regex and maybe collections for counting.2. Function definition: def most_frequent_word(s):3. Handle edge cases: what if the string is empty? Or has no words? Maybe return an empty string or handle it.4. Process the string: a. Convert to lowercase: s_lower = s.lower() b. Use re.findall(r'b[a-z]+b', s_lower) — wait, no, the word boundaries might not capture all cases. Or perhaps just find all sequences of letters, regardless of what's around them. So, re.findall(r'[a-z]+', s_lower). But wait, this would split on any non-letter, so for example, "don't" would become "dond" and "t", which is not correct. Hmm, but the problem statement says to ignore punctuation, so perhaps we should split on word boundaries and extract only the letters.Wait, maybe the correct approach is to split the string into tokens, considering words as sequences of letters, and ignoring any non-letter characters. So, using re.findall(r'b[a-z]+b', s_lower) might not be right because word boundaries can be tricky. Alternatively, perhaps the pattern [a-z]+ would capture all sequences of letters, regardless of what's around them.Wait, let's test this. For example, the string "Hello,world!" would be split into ["hello", "world"] using [a-z]+. That's correct. Similarly, "don't" would be split into ["don", "t"], which is not correct. But the problem statement says to ignore punctuation. So, perhaps in this case, apostrophes are considered part of the word. Hmm, but the problem statement isn't clear on that.Wait, the problem says to ignore punctuation. So, perhaps any punctuation attached to a word should be stripped. So, for example, "don't" is considered as "dont"? Or is the apostrophe kept, making it "don't"?Hmm, the example given doesn't have such cases. So perhaps, for the purpose of this problem, words are sequences of letters, and any non-letter character is treated as a word separator.So, perhaps the correct approach is to split the string into words by any non-letter character, and then collect the words, ignoring empty strings.So, in code:words = re.findall(r'[a-z]+', s_lower)Wait, but that would include words like 'a', 'i', etc., which are valid. So, that's acceptable.So, for the input string, after lowercasing, find all sequences of one or more letters.Once I have the list of words, I can count their occurrences.So, using a dictionary to count frequencies.Alternatively, using collections.Counter.So:from collections import Countercounts = Counter(words)Then, find the word with the maximum count.But what if there are multiple words with the same maximum count? The problem says to return any one of them.So, how to find the maximum.One approach is to get the most common element.But wait, Counter.most_common() returns a list of (word, count) pairs, ordered by count descending.So, the first element is the most frequent. If there are multiple with the same count, the first one in the list is returned.But wait, the order in which they appear in the list may depend on their order in the original string if counts are the same.But the problem says to return any one of them, so that's acceptable.So, the code would be:if not counts: return '' # or handle empty casemost_common = counts.most_common(1)[0][0]But wait, what if the input string has no words? Like, all punctuation or empty string. Then, counts would be empty, and trying to access most_common would throw an error.So, need to handle that.So, in code:if not words: return '' # or perhaps return None, but the problem says to return a word, so maybe return empty string.Wait, the note says the function should handle edge cases, like strings with only one word or no words. So, for no words, perhaps return an empty string.So, putting it all together.Let me outline the steps:1. Convert the input string to lowercase.2. Use regex to find all sequences of letters, resulting in a list of words.3. If the list is empty, return an empty string.4. Count the frequency of each word.5. Find the word with the highest frequency. If multiple, return any.So, the code:import refrom collections import Counterdef most_frequent_word(s): # Convert to lowercase s_lower = s.lower() # Find all words (sequences of letters) words = re.findall(r'[a-z]+', s_lower) if not words: return '' # Count frequencies counts = Counter(words) # Get the most common word most_common = counts.most_common(1)[0][0] return most_commonWait, but let's test this with the example.Sample input: "The quick brown fox jumps over the lazy dog. The dog barks loudly."After lowercasing: "the quick brown fox jumps over the lazy dog. the dog barks loudly."Regex finds: ['the', 'quick', 'brown', 'fox', 'jumps', 'over', 'the', 'lazy', 'dog', 'the', 'dog', 'barks', 'loudly']Counts:the: 3quick:1, brown:1, fox:1, jumps:1, over:1, lazy:1, dog:2, barks:1, loudly:1.So most_common is 'the', which is correct.Another test case: input is "Hello, hello! HELLO." The function should return 'hello'.Yes, because after lowercasing, the words are ['hello', 'hello', 'hello'].Another test case: input is "A a a a B b b ccc". The words are ['a', 'a', 'a', 'a', 'b', 'b', 'b', 'ccc']. So counts: a:4, b:3, ccc:1. So most common is 'a'.Another test case: input is "Hello world! Hello, world." The words are ['hello', 'world', 'hello', 'world']. So counts: hello:2, world:2. So the function returns 'hello' because it's the first in the list.Wait, but in Counter, the order is based on insertion. So in this case, 'hello' comes first in the list, so it's the first in the most_common list.But what if the words are in a different order? Like, if the string is "world hello world hello", then the counts are same, but the first word in the list is 'world', so the function returns 'world'.But the problem says to return any one of them, so that's acceptable.What about edge cases:Case 1: Empty string. The function returns empty string.Case 2: String with only punctuation. Like "!!! ???". The regex returns an empty list, so function returns empty string.Case 3: Single word. Returns that word.Another test case: input is "Don't worry." The regex would split into ['dont', 'worry'] because the apostrophe is not a letter. So counts are 1 each. So function returns 'dont' or 'worry' depending on which comes first.Wait, but in the string "Don't worry", the regex [a-z]+ would find 'dont' and 'worry' because the apostrophe is not a letter. So the word 'don't' is split into 'dont' and 't'? Wait, no. Let me think: the string is "Don't worry." After lowercasing, it's "don't worry."The regex [a-z]+ would find 'don', 't', 'worry'. Because the apostrophe is not a letter, so it's a separator. So the words are 'don', 't', 'worry'.So, in this case, the counts are 'don':1, 't':1, 'worry':1. So function returns 'don' as it's the first.But perhaps the problem expects 'don't' as a single word, but the problem statement says to ignore punctuation. So in this case, the apostrophe is considered punctuation and is ignored, so the word is split into 'don' and 't'.Hmm, but maybe the problem expects that apostrophes are part of the word. So perhaps the regex should include apostrophes as part of words.Wait, the problem statement says to ignore punctuation. So perhaps, any punctuation is stripped from the word, but the word itself is considered as a sequence of letters, regardless of punctuation.Wait, perhaps the correct approach is to remove all punctuation from the string before splitting into words.Alternatively, perhaps the regex should match word characters, including apostrophes.Wait, but the problem statement is a bit ambiguous. Let me re-read the note.The function should ignore case and punctuation, and return the most common word in lowercase.So, perhaps, the punctuation is stripped from the words. So, for example, "don't" becomes "dont", but perhaps the apostrophe is considered punctuation and is removed.Alternatively, perhaps the apostrophe is kept as part of the word.Wait, but the problem statement says to ignore punctuation. So, perhaps, any punctuation is stripped, so "don't" becomes "dont".But how to handle that in the regex.Hmm, perhaps the correct approach is to remove all non-alphanumeric characters except apostrophes, but that's complicating things.Alternatively, perhaps the regex should match any sequence of letters, ignoring any non-letters.So, in the example, "don't" would be split into 'don' and 't'.But perhaps the problem expects that the apostrophe is kept, so "don't" is considered as a single word.Wait, but the problem says to ignore punctuation. So perhaps, the apostrophe is considered punctuation and is stripped.But I'm not sure. The example given doesn't include such cases.Alternatively, perhaps the problem expects that words are sequences of letters, and any non-letter is treated as a word separator.So, in that case, the regex [a-z]+ is correct.So, in the case of "don't", it's split into 'don' and 't'.But perhaps, the problem expects that apostrophes are part of the word, so the regex should include apostrophes.So, perhaps the regex should be [a-z']+, but then, words like "don't" would be considered as a single word.But then, how to handle cases where apostrophes are at the beginning or end, like "'hello" or "hello'"?Hmm, perhaps the problem expects that apostrophes are part of the word.But the problem statement isn't clear on this.Wait, looking back at the problem statement: the function should ignore case and punctuation. So, perhaps, any punctuation is stripped, but letters are kept.So, perhaps, the correct approach is to remove all punctuation before processing.So, perhaps, the steps are:1. Convert to lowercase.2. Remove all punctuation from the string.3. Split into words.But how to remove punctuation.Alternatively, perhaps, the regex can be adjusted to include apostrophes as part of words.Wait, perhaps the regex should be [a-z']+, but then, words like "don't" are considered as one word.But then, what about words like "can't" — it would be considered as 'can't'.But then, in the counts, "can't" and "cant" would be considered different words.But the problem says to ignore punctuation, so perhaps, the apostrophe is considered punctuation and should be removed.So, perhaps, the correct approach is to remove all punctuation, including apostrophes, before processing.So, perhaps, the steps are:- Convert the string to lowercase.- Remove all punctuation (using a regex to replace all non-letters with spaces, then split into words).Wait, perhaps, the correct approach is to split the string into words by any non-letter character, and then collect the words.So, perhaps, the regex [a-z]+ is correct.So, in the case of "don't", it's split into 'don' and 't'.But perhaps, the problem expects that apostrophes are kept. So, perhaps, the regex should be [a-z']+, but then, the apostrophe is part of the word.But this is unclear.Alternatively, perhaps, the problem expects that words are sequences of letters, and any non-letter is treated as a separator.So, in that case, the regex [a-z]+ is correct.So, perhaps, the initial approach is correct.But perhaps, the problem expects that apostrophes are kept as part of the word.In that case, the regex should be [a-z']+, but then, we need to make sure that words are correctly formed.But perhaps, the problem doesn't have such test cases, and the initial approach is sufficient.So, proceeding with the initial code.Wait, but let's test the code with the example.In the example, the input is "The quick brown fox jumps over the lazy dog. The dog barks loudly."The code converts to lowercase, then finds all [a-z]+ sequences.So, the words are:['the', 'quick', 'brown', 'fox', 'jumps', 'over', 'the', 'lazy', 'dog', 'the', 'dog', 'barks', 'loudly']Counts: the:3, dog:2, others:1.So, the function returns 'the' which is correct.Another test case: input is "Hello, my name is Alice. Alice is my friend."The words are ['hello', 'my', 'name', 'is', 'alice', 'alice', 'is', 'my', 'friend']Counts: alice:2, my:2, is:2, others:1.So, the function returns 'alice' as it's the first in the list.But if the order is different, it may return 'my' or 'is'.But the problem allows any of them.So, the code seems to handle that.Another test case: input is "A a a a B b b ccc". The words are ['a','a','a','a','b','b','b','ccc'].Counts: a:4, b:3, ccc:1.So, function returns 'a'.Another test case: input is "Hello world! Hello, world." The words are ['hello','world','hello','world'].Counts: hello:2, world:2.Function returns 'hello' because it's the first in the list.But if the string is "world hello world hello", the function returns 'world'.So, the code is correct.Now, what about the edge case where the string is empty? The function returns empty string.Another edge case: string is " ... " — all punctuation. The function returns empty string.Another edge case: string is "Hi! Hi! Hi!" — the words are ['hi','hi','hi'], function returns 'hi'.So, the code seems to handle all these cases.Wait, but what about uppercase letters? The code converts to lowercase, so that's handled.So, I think the code is correct.But wait, the regex [a-z]+ may not capture words with apostrophes. For example, "don't" becomes 'don' and 't', which may not be intended.But the problem statement says to ignore punctuation, so perhaps the apostrophe is considered punctuation and is stripped.So, in that case, the code is correct.But perhaps, the problem expects that apostrophes are part of the word. So, perhaps, the regex should be adjusted.Wait, perhaps the problem expects that words are sequences of letters and apostrophes, but I'm not sure.But since the problem statement doesn't specify, perhaps the initial approach is acceptable.So, the code should be as written.

answer：Okay, so I have this problem about a rock formation shaped like a natural arch, and its width is modeled by a quadratic function. The function is given as ( W(x) = -0.5x^2 + 3x + 25 ), where ( x ) is the number of years since the arch was first measured. There are two parts to the problem. First, I need to determine when the width of the arch will reach its maximum value and what that maximum width is. Second, I have to find out the range of years during which the width remains above 15 meters, because the park authority wants to take action if it falls below that. Starting with the first part. Since the function is quadratic, and the coefficient of ( x^2 ) is negative (-0.5), the parabola opens downward. That means the vertex of the parabola will give me the maximum point. So, the vertex will tell me both the time when the width is maximum and the maximum width itself.I remember that for a quadratic function in the form ( ax^2 + bx + c ), the x-coordinate of the vertex is given by ( -b/(2a) ). So, in this case, ( a = -0.5 ) and ( b = 3 ). Plugging these into the formula, the x-coordinate (which represents the number of years) is ( -3/(2*(-0.5)) ). Let me compute that.First, compute the denominator: 2 times -0.5 is -1. So, it's -3 divided by -1, which is 3. So, the maximum width occurs at x = 3 years. Now, to find the maximum width, I need to plug x = 3 back into the original function ( W(x) ). Let's do that step by step.( W(3) = -0.5*(3)^2 + 3*(3) + 25 )Calculating each term:- ( (3)^2 = 9 )- ( -0.5 * 9 = -4.5 )- ( 3 * 3 = 9 )So, adding them all together: -4.5 + 9 + 25.Let me compute that: -4.5 + 9 is 4.5, and 4.5 + 25 is 29.5. So, the maximum width is 29.5 meters.Wait, that seems a bit high, but considering the quadratic model, it's possible. Let me double-check my calculations.Starting again:( W(3) = -0.5*(9) + 9 + 25 )- ( -0.5*9 = -4.5 )- Then, -4.5 + 9 = 4.5- 4.5 + 25 = 29.5Yes, that's correct. So, 29.5 meters is the maximum width, occurring at 3 years after the initial measurement.Moving on to the second part. The park authority wants to take action if the width falls below 15 meters. So, I need to find the range of years where the width is above 15 meters. Essentially, I need to solve the inequality ( W(x) > 15 ).So, let's set up the inequality:( -0.5x^2 + 3x + 25 > 15 )Subtract 15 from both sides to bring all terms to one side:( -0.5x^2 + 3x + 10 > 0 )Hmm, so now I have a quadratic inequality: ( -0.5x^2 + 3x + 10 > 0 ). To solve this, I should first find the roots of the corresponding quadratic equation ( -0.5x^2 + 3x + 10 = 0 ), and then determine the intervals where the quadratic expression is positive.But before solving, maybe I can make it a bit simpler by multiplying both sides by -2 to eliminate the decimal and the negative coefficient. However, I have to remember that multiplying both sides of an inequality by a negative number reverses the inequality sign.So, multiplying both sides by -2:( (-0.5x^2 + 3x + 10) * (-2) < 0 * (-2) )Which simplifies to:( x^2 - 6x - 20 < 0 )So now, the inequality is ( x^2 - 6x - 20 < 0 ). Let me find the roots of the equation ( x^2 - 6x - 20 = 0 ).Using the quadratic formula: ( x = [6 ± sqrt(36 + 80)] / 2 ). Wait, because the quadratic is ( x^2 - 6x - 20 ), so a = 1, b = -6, c = -20.So, discriminant ( D = b^2 - 4ac = (-6)^2 - 4*1*(-20) = 36 + 80 = 116 ).So, sqrt(116). Let me compute sqrt(116). 10^2 is 100, 11^2 is 121, so sqrt(116) is between 10 and 11. Let me compute it more precisely.116 divided by 4 is 29, so sqrt(116) = 2*sqrt(29). Since sqrt(29) is approximately 5.385, so 2*5.385 is approximately 10.77.So, the roots are:( x = [6 ± 10.77]/2 )Calculating both roots:First root: (6 + 10.77)/2 = 16.77/2 ≈ 8.385Second root: (6 - 10.77)/2 = (-4.77)/2 ≈ -2.385So, the roots are approximately x ≈ 8.385 and x ≈ -2.385.Since x represents the number of years since the initial measurement, negative time doesn't make sense in this context. So, we can ignore the negative root.Now, the quadratic ( x^2 - 6x - 20 ) is a parabola opening upwards (since the coefficient of ( x^2 ) is positive). So, it will be below zero between its two roots. But since one root is negative and the other is positive, the expression ( x^2 - 6x - 20 < 0 ) is true for x between -2.385 and 8.385.But since x can't be negative, the inequality ( x^2 - 6x - 20 < 0 ) holds for x between 0 and approximately 8.385 years.But wait, let me think again. The original inequality after multiplying by -2 was ( x^2 - 6x - 20 < 0 ). So, the expression is less than zero between the roots. Since one root is negative and the other is positive, the expression is negative from negative infinity to -2.385 and from 8.385 to positive infinity? Wait, no, that's not right.Wait, no. For a parabola opening upwards, it is below zero between its two roots. So, if the roots are at x ≈ -2.385 and x ≈ 8.385, then the expression is negative between -2.385 and 8.385. So, in the context of our problem, x is greater than or equal to 0, so the expression is negative from x = 0 to x ≈ 8.385.But wait, hold on. The original inequality after multiplying by -2 was ( x^2 - 6x - 20 < 0 ), which is equivalent to ( -0.5x^2 + 3x + 10 > 0 ). So, the expression ( -0.5x^2 + 3x + 10 ) is greater than zero when ( x^2 - 6x - 20 < 0 ), which is between the roots.But since x cannot be negative, the interval where the width is above 15 meters is from x = 0 to x ≈ 8.385 years.Wait, but let me verify this because sometimes when dealing with inequalities, especially after multiplying both sides by a negative, it's easy to get confused.Alternatively, maybe I should solve the original inequality without multiplying both sides by -2.Original inequality: ( -0.5x^2 + 3x + 10 > 0 )Let me rewrite this as ( -0.5x^2 + 3x + 10 > 0 ). To make it easier, I can factor out the -0.5:( -0.5(x^2 - 6x - 20) > 0 )Divide both sides by -0.5, remembering to reverse the inequality:( x^2 - 6x - 20 < 0 )Which is the same as before. So, the solution is x between the roots, which are approximately -2.385 and 8.385. Since x must be non-negative, the solution is 0 ≤ x < 8.385.But let me check the value at x = 0: ( W(0) = -0.5*(0)^2 + 3*(0) + 25 = 25 ), which is above 15. At x = 8.385, the width is 15 meters. So, the width is above 15 meters from year 0 up until approximately 8.385 years.But to express this more precisely, maybe I should find the exact roots instead of approximate.So, going back to the quadratic equation ( x^2 - 6x - 20 = 0 ). The exact roots are:( x = [6 ± sqrt(36 + 80)] / 2 = [6 ± sqrt(116)] / 2 )Simplify sqrt(116): sqrt(4*29) = 2*sqrt(29). So, the roots are:( x = [6 ± 2sqrt(29)] / 2 = 3 ± sqrt(29) )So, the exact roots are ( x = 3 + sqrt(29) ) and ( x = 3 - sqrt(29) ). Since sqrt(29) is approximately 5.385, so 3 + 5.385 ≈ 8.385 and 3 - 5.385 ≈ -2.385, which matches our earlier approximation.Therefore, the exact interval where the width is above 15 meters is from x = 0 to x = 3 + sqrt(29). Since 3 + sqrt(29) is approximately 8.385, the width remains above 15 meters until about 8.385 years.But the problem asks for the range of years during which the width remains above 15 meters. So, it's from year 0 up to year 3 + sqrt(29). But since the park authority is monitoring from the initial measurement, which is x = 0, the range is 0 ≤ x < 3 + sqrt(29). But maybe they want it in terms of the number of years, so it's from year 0 to approximately 8.385 years. But since we can't have a fraction of a year in the context, perhaps we need to consider whole years? Or maybe they just want the exact expression.Wait, the problem says "the range of years during which the width of the arch remains above 15 meters." It doesn't specify whether to round or give an exact value. So, probably better to give the exact value in terms of sqrt(29). So, the width is above 15 meters from year 0 to year 3 + sqrt(29).But let me check if at x = 3 + sqrt(29), the width is exactly 15 meters. Plugging back into the original function:( W(3 + sqrt(29)) = -0.5*(3 + sqrt(29))^2 + 3*(3 + sqrt(29)) + 25 )Let me compute this step by step.First, compute ( (3 + sqrt(29))^2 ):= 9 + 6sqrt(29) + 29= 38 + 6sqrt(29)So, ( -0.5*(38 + 6sqrt(29)) = -19 - 3sqrt(29) )Next, compute 3*(3 + sqrt(29)) = 9 + 3sqrt(29)Adding all terms together:-19 - 3sqrt(29) + 9 + 3sqrt(29) + 25Simplify:-19 + 9 + 25 = 15-3sqrt(29) + 3sqrt(29) = 0So, yes, it equals 15. So, at x = 3 + sqrt(29), the width is exactly 15 meters. Therefore, the width is above 15 meters for x between 0 and 3 + sqrt(29). But since x is measured in years, and we can't have a fraction of a year in this context, maybe we need to consider the integer years. But the problem doesn't specify, so perhaps we can leave it as an exact value.Alternatively, if they want the range in terms of years, it's from year 0 to approximately 8.385 years, so about 8.39 years. But since the problem is mathematical, exact form is probably preferred.So, summarizing:1. The maximum width occurs at x = 3 years, and the maximum width is 29.5 meters.2. The width remains above 15 meters from year 0 to year 3 + sqrt(29), which is approximately 8.385 years.I think that's it. Let me just recap to make sure I didn't make any mistakes.For part 1, vertex at x = -b/(2a) = -3/(2*(-0.5)) = 3. Plugging back in, W(3) = 29.5. That seems correct.For part 2, solving ( W(x) > 15 ), which led to x between 0 and 3 + sqrt(29). That seems correct as well.I think I'm confident with these answers.**Final Answer**1. The width of the arch reaches its maximum value after boxed{3} years, with a maximum width of boxed{29.5} meters.2. The width of the arch remains above 15 meters from year boxed{0} to year boxed{3 + sqrt{29}}.

answer：Okay, so I have this problem about a local resident who loves collecting unique and antique artifacts. They go to this store called Eastern Elements and find a vase and a clock. The vase costs x and the clock costs y. The store has a promotion: if the total price before discount exceeds 500, you get a 10% discount. There are two parts to this problem. Let me tackle them one by one.**Problem 1: Find the expression for the total price before the discount.**The vase's price is given by the polynomial (2x^2 - 3x + 250), and the clock's price is (3y^2 + 4y + 150). So, to find the total price before discount, I just need to add these two expressions together.Let me write that out:Total price = Price of vase + Price of clockTotal price = ( (2x^2 - 3x + 250) + (3y^2 + 4y + 150) )Now, I should combine like terms. Let's see, the (x^2) term is only in the vase's price, so that's (2x^2). Similarly, the (y^2) term is only in the clock's price, so that's (3y^2). Next, the linear terms: the vase has (-3x) and the clock has (+4y). So, we have (-3x + 4y). Finally, the constant terms: 250 from the vase and 150 from the clock. Adding those together gives 400.Putting it all together, the total price expression is:(2x^2 + 3y^2 - 3x + 4y + 400)Hmm, that seems right. Let me double-check:- Vase: (2x^2 - 3x + 250)- Clock: (3y^2 + 4y + 150)- Adding them: (2x^2 + 3y^2 - 3x + 4y + (250 + 150))- Which simplifies to (2x^2 + 3y^2 - 3x + 4y + 400)Yep, that looks correct.**Problem 2: Determine the values of (x) and (y) such that the total price exceeds 500, and then calculate the discounted price using these values. Assume (x) and (y) are integers, and (x = y + 10).**Alright, so we need to find integer values of (x) and (y) where (x = y + 10), and the total price before discount is more than 500. Then, calculate the discounted price.First, let's substitute (x = y + 10) into the total price expression. That way, we can express everything in terms of (y) and solve for (y).Given that the total price is (2x^2 + 3y^2 - 3x + 4y + 400), substituting (x = y + 10):Let me compute each term step by step.1. (2x^2 = 2(y + 10)^2)2. (3y^2) remains as is.3. (-3x = -3(y + 10))4. (4y) remains as is.5. The constant term is 400.So, let's expand each term:1. (2(y + 10)^2 = 2(y^2 + 20y + 100) = 2y^2 + 40y + 200)2. (3y^2) is just (3y^2)3. (-3(y + 10) = -3y - 30)4. (4y) is (4y)5. 400 remains 400.Now, let's combine all these:Total price = (2y^2 + 40y + 200 + 3y^2 - 3y - 30 + 4y + 400)Combine like terms:- (2y^2 + 3y^2 = 5y^2)- (40y - 3y + 4y = 41y)- (200 - 30 + 400 = 570)So, the total price in terms of (y) is:(5y^2 + 41y + 570)Now, we need this total price to exceed 500:(5y^2 + 41y + 570 > 500)Subtract 500 from both sides:(5y^2 + 41y + 70 > 0)Hmm, so we have a quadratic inequality here: (5y^2 + 41y + 70 > 0). Let me solve this inequality.First, let's find the roots of the quadratic equation (5y^2 + 41y + 70 = 0).Using the quadratic formula:(y = frac{-b pm sqrt{b^2 - 4ac}}{2a})Where (a = 5), (b = 41), and (c = 70).Compute discriminant:(D = b^2 - 4ac = 41^2 - 4*5*70 = 1681 - 1400 = 281)So, the roots are:(y = frac{-41 pm sqrt{281}}{10})Compute approximate values:(sqrt{281}) is approximately 16.763.So,(y = frac{-41 + 16.763}{10} = frac{-24.237}{10} ≈ -2.4237)and(y = frac{-41 - 16.763}{10} = frac{-57.763}{10} ≈ -5.7763)So, the quadratic crosses the y-axis at approximately y ≈ -2.4237 and y ≈ -5.7763.Since the coefficient of (y^2) is positive (5), the parabola opens upwards. Therefore, the quadratic is positive outside the interval between the roots.So, the inequality (5y^2 + 41y + 70 > 0) holds when (y < -5.7763) or (y > -2.4237).But, since (y) is an integer, and (x = y + 10) must also be an integer, we need to find integer values of (y) such that either (y < -5.7763) or (y > -2.4237).But wait, let's think about this. The price expressions for the vase and clock are polynomials in (x) and (y). Are there any constraints on (x) and (y)? For example, prices can't be negative, so (2x^2 - 3x + 250 > 0) and (3y^2 + 4y + 150 > 0). Let me check if these polynomials are always positive.For the vase: (2x^2 - 3x + 250). The discriminant is (9 - 2000 = -1991), which is negative, so this quadratic is always positive. Similarly, for the clock: (3y^2 + 4y + 150). Discriminant is (16 - 1800 = -1784), also negative. So both are always positive, regardless of (x) and (y). So, (x) and (y) can be any integers, positive or negative, but in reality, prices can't be negative, so (x) and (y) must be such that the polynomials evaluate to positive numbers, but since they are always positive, any integer (x) and (y) is acceptable.But, in the context of the problem, (x) and (y) are likely positive integers because they represent prices. So, (x) and (y) should be positive integers.Given that (x = y + 10), and both (x) and (y) are positive integers, (y) must be at least 1, making (x = 11).Wait, but earlier, the inequality (5y^2 + 41y + 70 > 0) is satisfied for (y < -5.7763) or (y > -2.4237). But since (y) is a positive integer, (y > -2.4237) is always true. So, the inequality is automatically satisfied for all positive integers (y). Wait, hold on. Let me compute the total price for (y = 1):Total price = (5(1)^2 + 41(1) + 570 = 5 + 41 + 570 = 616). Which is greater than 500. So, for any positive integer (y), the total price will be greater than 500.But wait, let me check (y = 0):Total price = (5(0)^2 + 41(0) + 570 = 570), which is still greater than 500. So, even for (y = 0), the total price is 570.But (y = 0) would make (x = y + 10 = 10). Let's check the prices:Vase price: (2(10)^2 - 3(10) + 250 = 200 - 30 + 250 = 420)Clock price: (3(0)^2 + 4(0) + 150 = 0 + 0 + 150 = 150)Total: 420 + 150 = 570, which is correct.So, actually, for (y = 0), the total is 570, which is above 500.But if (y = -1), let's see:Total price = (5(-1)^2 + 41(-1) + 570 = 5 - 41 + 570 = 534). Still above 500.Wait, but (y = -1) would make (x = y + 10 = 9). Let's compute the prices:Vase: (2(9)^2 - 3(9) + 250 = 162 - 27 + 250 = 385)Clock: (3(-1)^2 + 4(-1) + 150 = 3 - 4 + 150 = 149)Total: 385 + 149 = 534, which is correct.So, even for negative (y), as long as (x = y + 10) is positive, the total price is above 500.But in the problem statement, it says (x) and (y) are integers. It doesn't specify they have to be positive. So, technically, (y) can be any integer, positive or negative, as long as (x = y + 10) is also an integer, which it will be since (y) is integer.But, in reality, prices can't be negative. So, we need to ensure that both the vase and clock prices are positive. Since we already saw that the polynomials for vase and clock are always positive regardless of (x) and (y), so even if (x) or (y) are negative, the prices are still positive.But, in the context of the problem, the resident is visiting a store, so it's more realistic that (x) and (y) are positive integers. So, perhaps we should consider (y) as a positive integer.But the problem doesn't specify, so maybe we should consider all integer values where (x = y + 10), regardless of sign.But given that the total price is always above 500 for (y) being any integer, as we saw even for (y = -10):Total price = (5(-10)^2 + 41(-10) + 570 = 500 - 410 + 570 = 660), which is above 500.Wait, but if (y = -11):Total price = (5(-11)^2 + 41(-11) + 570 = 605 - 451 + 570 = 724), still above 500.Wait, but if (y) is a very large negative number, say (y = -100):Total price = (5(10000) + 41(-100) + 570 = 50000 - 4100 + 570 = 46470), which is way above 500.Wait, but actually, as (y) becomes more negative, the total price increases because the (5y^2) term dominates, which is positive.So, actually, for all integer values of (y), the total price is above 500. Because even when (y) is negative, the (5y^2) term is positive and large enough to make the total price exceed 500.Wait, but let's test (y = -6):Total price = (5(36) + 41(-6) + 570 = 180 - 246 + 570 = 504). So, 504 is just above 500.Wait, so for (y = -6), total price is 504, which is just above 500.But if (y = -7):Total price = (5(49) + 41(-7) + 570 = 245 - 287 + 570 = 528), which is also above 500.Wait, but when (y = -6), total is 504, which is just over 500. So, what about (y = -5):Total price = (5(25) + 41(-5) + 570 = 125 - 205 + 570 = 490). That's below 500.Wait, so for (y = -5), total price is 490, which is below 500.Similarly, (y = -4):Total price = (5(16) + 41(-4) + 570 = 80 - 164 + 570 = 486). Also below 500.Wait, so the total price is above 500 when (y leq -6) or (y geq -2). Because for (y = -6), it's 504, which is above 500, and for (y = -5), it's 490, which is below. Similarly, for (y = -2), let's compute:Total price = (5(4) + 41(-2) + 570 = 20 - 82 + 570 = 508). So, 508 is above 500.For (y = -3):Total price = (5(9) + 41(-3) + 570 = 45 - 123 + 570 = 492). Below 500.So, the total price exceeds 500 when (y leq -6) or (y geq -2). But since (y) is an integer, we can say (y leq -6) or (y geq -2).But in the context of the problem, (y) is likely a positive integer because it's the price of a clock, which can't be negative. So, (y) should be a positive integer, which would make (x = y + 10) also a positive integer.But just to be thorough, let's consider both cases.Case 1: (y) is a positive integer.Then, (x = y + 10) is also a positive integer, and the total price is (5y^2 + 41y + 570), which is definitely above 500 for any positive (y).Case 2: (y) is a negative integer.But in this case, (x = y + 10) must also be an integer. If (y) is negative, (x) could be positive or negative depending on how negative (y) is.But as we saw earlier, for (y leq -6), the total price is above 500, but for (y = -5, -4, -3, -2, -1, 0), the total price is sometimes above and sometimes below 500.Wait, actually, for (y = -2), total price is 508, which is above 500, but (y = -3) gives 492, which is below.So, to have the total price exceed 500, (y) must satisfy (y leq -6) or (y geq -2). But since (y) is an integer, (y) can be any integer less than or equal to -6 or greater than or equal to -2.But in the context of the problem, (y) is the price of a clock, so it's more realistic that (y) is a positive integer. So, we can focus on positive integers for (y).Therefore, for any positive integer (y), the total price will exceed 500, and the resident will get a 10% discount.But the problem says "determine the values of (x) and (y)" such that total exceeds 500. It doesn't specify a particular total or a minimal total, so technically, there are infinitely many solutions where (x = y + 10) and (y) is an integer (positive or negative as per the constraints above). But since the problem says "determine the values", it might be expecting specific values, but it's not clear.Wait, perhaps I misread. Let me check the problem again."Determine the values of (x) and (y) such that the total price exceeds 500, and then calculate the discounted price using these values. Assume (x) and (y) are integers, and (x = y + 10)."So, it's asking for values of (x) and (y) (plural), so maybe all possible integer pairs where (x = y + 10) and the total exceeds 500. But since there are infinitely many, perhaps it's expecting a general expression or maybe the minimal values.Wait, but the problem doesn't specify minimal values or anything. It just says "determine the values". Maybe it's expecting us to express (x) and (y) in terms of each other, but since (x = y + 10), and the total price is a quadratic in (y), which is always above 500 for (y geq -2) or (y leq -6). So, the solution set is all integers (y) such that (y geq -2) or (y leq -6), with (x = y + 10).But since the problem is presented in a math problem context, maybe it's expecting specific numerical values. But without more constraints, we can't determine specific values. So, perhaps the problem is expecting us to express the discounted price in terms of (y), or maybe find a general expression.Wait, let me reread the problem:"Determine the values of (x) and (y) such that the total price exceeds 500, and then calculate the discounted price using these values. Assume (x) and (y) are integers, and (x = y + 10)."Hmm, so it's saying "determine the values" (plural), so maybe all possible integer pairs (x, y) where (x = y + 10) and total > 500. But as we saw, that's an infinite set. So, perhaps the problem is expecting to express the discounted price in terms of (y), given that (x = y + 10).Alternatively, maybe it's expecting to find the minimal (y) such that the total exceeds 500, but in that case, for positive (y), the minimal (y) is 1, since (y = 0) gives total 570, which is above 500.Wait, but (y = 0) is allowed? If (y = 0), then the clock's price is (3(0)^2 + 4(0) + 150 = 150), which is fine. The vase's price would be (2(10)^2 - 3(10) + 250 = 420). So, total is 570, which is above 500.So, the minimal positive integer (y) is 0, but since (y) is an integer, and 0 is an integer, that's acceptable.But in the context of the problem, maybe (y) should be at least 1, as a price of 0 for a clock doesn't make sense. So, if we consider (y geq 1), then the minimal (y) is 1, giving (x = 11), total price 616, discounted price 616 - 10% = 554.4.But the problem doesn't specify minimal or anything, so perhaps it's expecting a general expression for the discounted price in terms of (y), given that (x = y + 10).Wait, let's see:Total price is (5y^2 + 41y + 570). If total exceeds 500, which it does for (y geq -2) or (y leq -6), then the discounted price is 90% of the total.So, discounted price = 0.9 * (5y^2 + 41y + 570)But maybe we can write it as:Discounted price = (4.5y^2 + 36.9y + 513)But since the problem mentions "calculate the discounted price using these values", it might be expecting specific numerical values. But without specific (x) and (y), we can't compute a numerical value. So, perhaps the problem is expecting an expression in terms of (y), or maybe to express the discounted price as a function of (y).Alternatively, maybe the problem is expecting to find the minimal (y) such that the total exceeds 500, and then compute the discounted price for that minimal (y). If we consider (y) as a positive integer, the minimal (y) is 0, but if we consider (y) as a positive integer greater than 0, then minimal (y) is 1.Let me compute the discounted price for (y = 0):Total price = 570Discounted price = 570 - 10% of 570 = 570 - 57 = 513For (y = 1):Total price = 616Discounted price = 616 - 61.6 = 554.4But since the problem says "calculate the discounted price using these values", and it's asking for values of (x) and (y), it's unclear whether it wants a general expression or specific values.Wait, perhaps the problem is expecting to express the discounted price in terms of (y), given (x = y + 10). So, the discounted price is 0.9*(5y^2 + 41y + 570) = 4.5y^2 + 36.9y + 513. But since (x) and (y) are integers, and the discounted price would be a decimal, but the problem doesn't specify rounding or anything.Alternatively, maybe the problem expects us to leave it as 90% of the total, expressed as (0.9(5y^2 + 41y + 570)).But I'm not sure. Let me think again.The problem says:"Determine the values of (x) and (y) such that the total price exceeds 500, and then calculate the discounted price using these values. Assume (x) and (y) are integers, and (x = y + 10)."So, it's two parts:1. Find (x) and (y) such that total > 500.2. Using these values, calculate the discounted price.But since there are infinitely many such (x) and (y), unless it's expecting a general expression, or perhaps the minimal (y).Wait, maybe it's expecting to express the discounted price in terms of (y), given (x = y + 10). So, the discounted price is 0.9*(5y^2 + 41y + 570). So, that's an expression.Alternatively, maybe it's expecting to write the discounted price as a polynomial in (y), which would be (4.5y^2 + 36.9y + 513). But since the problem mentions "calculate the discounted price", it might be expecting a numerical value, but without specific (y), we can't compute a number.Wait, perhaps the problem is expecting to find the minimal (y) such that the total exceeds 500, and then compute the discounted price for that minimal (y). So, if (y) is a positive integer, the minimal (y) is 0, but if (y) must be positive, then (y = 1).But let's check for (y = 0):Total price = 570Discounted price = 570 * 0.9 = 513For (y = 1):Total price = 616Discounted price = 616 * 0.9 = 554.4But the problem doesn't specify minimal or anything, so maybe it's expecting to express the discounted price in terms of (y). So, the discounted price is (0.9(5y^2 + 41y + 570)), which simplifies to (4.5y^2 + 36.9y + 513).But since the problem mentions "calculate the discounted price using these values", and "these values" refer to the values of (x) and (y) that make the total exceed 500, which are infinitely many, perhaps the answer is just the expression for the discounted price.Alternatively, maybe the problem is expecting to express the discounted price as a function of (y), given (x = y + 10), which we have as (4.5y^2 + 36.9y + 513).But I'm not sure. Maybe I should present both the expression for the total price and the discounted price in terms of (y).Wait, but the problem is in two parts:1. Find the total price expression.2. Determine (x) and (y) such that total > 500, then calculate discounted price.So, for part 2, since (x = y + 10), and the total is (5y^2 + 41y + 570), which is always above 500 for (y geq -2) or (y leq -6). So, the values of (x) and (y) are all integers where (y geq -2) or (y leq -6), with (x = y + 10).Then, the discounted price is 90% of the total, which is (0.9(5y^2 + 41y + 570)).But since the problem says "calculate the discounted price using these values", and it's plural, maybe it's expecting to express the discounted price in terms of (y), which is (4.5y^2 + 36.9y + 513).Alternatively, if we consider (y) as a positive integer, then the discounted price is (4.5y^2 + 36.9y + 513).But let me check if the problem expects specific numerical values. Since it's part 2, and part 1 was an expression, part 2 might be expecting an expression as well.Alternatively, maybe the problem is expecting to find the minimal (y) such that the total exceeds 500, and then compute the discounted price for that minimal (y). If (y) is a positive integer, the minimal (y) is 0, but if (y) must be positive, then (y = 1). Let's compute both.For (y = 0):Total price = 570Discounted price = 570 * 0.9 = 513For (y = 1):Total price = 616Discounted price = 616 * 0.9 = 554.4But since (y = 0) is allowed, and gives a lower discounted price, maybe that's the answer.But the problem doesn't specify minimal or anything, so perhaps it's expecting to express the discounted price in terms of (y).Alternatively, maybe the problem is expecting to write the discounted price as a polynomial in (y), which is (4.5y^2 + 36.9y + 513).But since the problem mentions "calculate the discounted price using these values", and "these values" refer to (x) and (y), which are related by (x = y + 10), perhaps the answer is the expression (4.5y^2 + 36.9y + 513).But I'm not entirely sure. Maybe I should present both the total price expression and the discounted price expression.Wait, but the problem is in two parts:1. Find the total price expression.2. Determine (x) and (y) such that total > 500, then calculate discounted price.So, for part 2, the answer would be:All integer pairs ((x, y)) where (x = y + 10) and (y geq -2) or (y leq -6). The discounted price for these values is (0.9(5y^2 + 41y + 570)), which simplifies to (4.5y^2 + 36.9y + 513).But since the problem says "calculate the discounted price using these values", it might be expecting a numerical answer, but without specific (y), we can't compute a numerical value. So, perhaps the answer is the expression (4.5y^2 + 36.9y + 513).Alternatively, if the problem expects specific values, maybe it's the minimal (y), which is (y = 0), giving discounted price 513.But I'm not sure. Maybe I should go with the expression.So, to summarize:1. Total price expression: (2x^2 + 3y^2 - 3x + 4y + 400)2. Values of (x) and (y) are integers with (x = y + 10) and (y geq -2) or (y leq -6). The discounted price is (4.5y^2 + 36.9y + 513).But let me check if the problem expects the discounted price to be in a specific form. Since the original prices are polynomials with integer coefficients, maybe the discounted price should be expressed as a polynomial with decimal coefficients, which is what I have.Alternatively, maybe factor out the 0.9:Discounted price = (0.9(5y^2 + 41y + 570))But I think expressing it as (4.5y^2 + 36.9y + 513) is fine.Wait, but in the problem, the prices are given as polynomials with integer coefficients, so maybe the discounted price should also be expressed with fractions instead of decimals.Let me compute 0.9*(5y^2 + 41y + 570):0.9 = 9/10, so:Discounted price = (9/10)(5y^2 + 41y + 570) = (45/10)y^2 + (369/10)y + (5130/10) = (9/2)y^2 + (369/10)y + 513But that's still not all integers. Alternatively, we can write it as:Discounted price = (frac{9}{10}(5y^2 + 41y + 570))But that might be the simplest form.Alternatively, factor out 9/10:Discounted price = (frac{9}{10}(5y^2 + 41y + 570))But I think expressing it as (4.5y^2 + 36.9y + 513) is acceptable.Alternatively, if we want to write it with fractions:4.5 = 9/2, 36.9 = 369/10, 513 = 513/1So, Discounted price = (frac{9}{2}y^2 + frac{369}{10}y + 513)But that's more complicated.Alternatively, leave it as (0.9(5y^2 + 41y + 570)), which is concise.But I think the problem expects the discounted price expressed as a polynomial, so (4.5y^2 + 36.9y + 513).Alternatively, if we multiply through:0.9*(5y^2) = 4.5y^20.9*(41y) = 36.9y0.9*570 = 513So, yes, that's correct.So, putting it all together:1. Total price expression: (2x^2 + 3y^2 - 3x + 4y + 400)2. Values of (x) and (y) are integers with (x = y + 10) and (y geq -2) or (y leq -6). The discounted price is (4.5y^2 + 36.9y + 513).But since the problem says "calculate the discounted price using these values", and "these values" refer to (x) and (y), which are related by (x = y + 10), perhaps the answer is the expression (4.5y^2 + 36.9y + 513).Alternatively, if the problem expects specific numerical values, maybe it's the minimal (y), which is (y = 0), giving discounted price 513.But I'm not sure. Maybe I should present both.Wait, but the problem is in two parts. Part 1 is an expression, part 2 is to determine values and calculate discounted price. So, for part 2, the answer would be the set of all integer pairs ((x, y)) where (x = y + 10) and (y geq -2) or (y leq -6), and the discounted price is (4.5y^2 + 36.9y + 513).But since the problem is likely expecting a specific answer, maybe it's the expression for the discounted price.Alternatively, maybe the problem is expecting to find the minimal (y) such that the total exceeds 500, which is (y = 0), and then compute the discounted price as 513.But I'm not sure. Maybe I should go with the expression.So, final answers:1. Total price expression: (2x^2 + 3y^2 - 3x + 4y + 400)2. Discounted price expression: (4.5y^2 + 36.9y + 513)But let me check if I can write it as a polynomial with fractions:4.5 = 9/2, 36.9 = 369/10, 513 = 513/1So, Discounted price = (frac{9}{2}y^2 + frac{369}{10}y + 513)Alternatively, factor out 9/10:Discounted price = (frac{9}{10}(5y^2 + 41y + 570))But I think the simplest is (4.5y^2 + 36.9y + 513).Alternatively, if we want to write it as a single fraction:Multiply all terms by 10 to eliminate decimals:45y^2 + 369y + 5130Then factor out 9:9(5y^2 + 41y + 570)But that's the same as 9/10*(5y^2 + 41y + 570)*10, which is not helpful.Alternatively, leave it as is.I think the answer is expected to be the expression (4.5y^2 + 36.9y + 513).But to confirm, let's compute for (y = 0):Discounted price = 4.5(0)^2 + 36.9(0) + 513 = 513, which matches.For (y = 1):4.5(1) + 36.9(1) + 513 = 4.5 + 36.9 + 513 = 554.4, which matches.So, the expression is correct.Therefore, the final answers are:1. Total price: (2x^2 + 3y^2 - 3x + 4y + 400)2. Discounted price: (4.5y^2 + 36.9y + 513)But since the problem mentions "calculate the discounted price using these values", and "these values" are (x) and (y), which are related by (x = y + 10), perhaps the answer is the expression (4.5y^2 + 36.9y + 513).Alternatively, if the problem expects specific numerical values, maybe it's the minimal (y), which is (y = 0), giving discounted price 513.But I think the answer is expected to be the expression.So, to write the final answer:1. The total price before discount is (2x^2 + 3y^2 - 3x + 4y + 400).2. The discounted price is (4.5y^2 + 36.9y + 513).But since the problem says "calculate the discounted price using these values", and "these values" are (x) and (y), which are related by (x = y + 10), perhaps the answer is the expression (4.5y^2 + 36.9y + 513).Alternatively, if the problem expects specific numerical values, maybe it's the minimal (y), which is (y = 0), giving discounted price 513.But I'm not sure. Maybe I should present both.Wait, but the problem is in two parts. Part 1 is an expression, part 2 is to determine values and calculate discounted price. So, for part 2, the answer would be the set of all integer pairs ((x, y)) where (x = y + 10) and (y geq -2) or (y leq -6), and the discounted price is (4.5y^2 + 36.9y + 513).But since the problem is likely expecting a specific answer, maybe it's the expression for the discounted price.Alternatively, maybe the problem is expecting to find the minimal (y) such that the total exceeds 500, which is (y = 0), and then compute the discounted price as 513.But I'm not sure. Maybe I should go with the expression.So, final answers:1. Total price expression: (2x^2 + 3y^2 - 3x + 4y + 400)2. Discounted price expression: (4.5y^2 + 36.9y + 513)But to present them boxed:1. boxed{2x^2 + 3y^2 - 3x + 4y + 400}2. boxed{4.5y^2 + 36.9y + 513}Alternatively, if the problem expects specific numerical values, maybe it's the minimal (y), which is (y = 0), giving discounted price 513.But I think the answer is expected to be the expression.So, I'll go with the expressions.

answer：Okay, I need to write a Python function that analyzes a list of movie reviews and returns the percentage of reviews containing the word "devil". Hmm, let's think about how to approach this.First, the function will take a list of strings as input. Each string is a review. So, for each review in the list, I need to check if the word "devil" is present.Wait, but how do I check if a word is present in a string? Oh right, I can use the 'in' keyword. So for each review, I'll check if 'devil' is in it.But wait, what about case sensitivity? Like, if the review has "Devil" with a capital D, will it count? The problem statement says to look for the word "devil", so I think it's case-sensitive. So I don't need to convert the reviews to lowercase or anything. Unless the problem expects it, but the example shows "devil" in lowercase, and the expected output counts it, so I think it's case-sensitive.So, the steps are:1. Initialize a counter to 0.2. Loop through each review in the input list.3. For each review, check if 'devil' is a substring.4. If it is, increment the counter.5. After processing all reviews, calculate the percentage by dividing the counter by the total number of reviews, then multiply by 100.6. Return this percentage as a float.Wait, what if the list is empty? Then dividing by zero would be a problem. But I think the function can assume that the input list is non-empty, as per the problem statement. Or perhaps, in code, we should handle that case to avoid division by zero. But the problem says it's a list of strings, so maybe it's safe to assume it's non-empty.Let me think about the example given. There are 5 reviews, 3 of which contain 'devil'. So 3/5 is 0.6, multiplied by 100 is 60.0, which matches the expected output.So, in code:def calculate_devil_percentage(reviews): count = 0 for review in reviews: if 'devil' in review: count +=1 percentage = (count / len(reviews)) * 100 return percentageWait, but what if the list is empty? Then len(reviews) is zero, and division by zero error occurs. So perhaps, the function should handle that case. But the problem says the function takes a list of strings as input, but doesn't specify if it's empty. So perhaps, we should add a condition to return 0.0 if the list is empty to avoid division by zero.But looking back at the problem statement's example, the input is a non-empty list. So maybe the function can assume that the input list is non-empty. Or perhaps, the function should return 0.0 if there are no reviews.Hmm, but in the problem statement, the output is a float, so 0.0 is acceptable. So perhaps, in code, I should handle the case when len(reviews) is zero to avoid division by zero.So, modifying the code:if len(reviews) == 0: return 0.0else: calculate as before.Alternatively, in Python, you can compute count / len(reviews) but if len is zero, it will throw an error. So, to prevent that, the function should handle the empty list case.So, putting it all together.Wait, but in the example, the function returns 60.0, which is a float. So the function should return a float, not an integer.So, the code seems solid.Testing the example:reviews = [ "This movie is devilishly good!", "I found the plot quite boring.", "The devil is in the details, and this film nails it.", "An average movie with nothing special.", "The protagonist's performance was devilishly charming."]Count is 3, len is 5. 3/5 is 0.6, 0.6*100 is 60.0.Yes, that's correct.Another test case: empty list. The function should return 0.0.Another test case: one review that has 'devil' → 100.0.Another test case: two reviews, one has 'devil', the other doesn't → 50.0.So the code should handle these.So, putting it all together, the function is as follows.