answer：Okay, I need to write a Python function that extracts and counts unique words from song lyrics. The input is a string, and the output should be a dictionary with words as keys and their counts as values. Hmm, let's think about how to approach this.First, I remember that words are separated by spaces, but there are also punctuation marks. So I need to split the string into words, but also remove any punctuation attached to them. Oh right, and the function should be case-insensitive, so all words should be converted to lowercase before counting.Wait, so how do I handle the punctuation? Like, if a word ends with a comma or exclamation point, I need to strip those off. Maybe I can iterate through each word and remove any non-alphanumeric characters from the start and end.Alternatively, perhaps using regular expressions would be a better approach. Oh right, regular expressions can help split the string into words, ignoring the punctuation. Or maybe find all the words, considering apostrophes or other characters as part of the word?Wait, the problem says to ignore punctuation marks when counting. So for example, in the sample input, "Hello," becomes "hello" and is counted once, but since it appears twice, the count is 2.So the plan is:1. Convert the entire lyrics string to lowercase to handle case insensitivity.2. Split the string into words, but also remove any punctuation attached to the words.3. For each word, strip any leading and trailing punctuation.4. If a word is empty after stripping, skip it.5. Count each word's occurrences.Alternatively, using regular expressions to find all the words, considering words as sequences of letters, possibly including apostrophes or other characters? Wait, the problem says to ignore punctuation, so perhaps any non-word character should be treated as a word separator.Wait, perhaps using the re.findall() function with a pattern that matches word characters. But then, what about apostrophes in words like "don't"? The problem statement doesn't specify, but the example doesn't have such cases. Hmm, but in the sample, the word "you!" becomes "you", so punctuation is stripped.So perhaps the approach is to split the lyrics into tokens, then for each token, remove any leading and trailing punctuation, and then check if the resulting string is non-empty.So step by step:- Split the lyrics into tokens based on whitespace. So each token is a word possibly with punctuation attached.- For each token, process it to remove leading and trailing punctuation.- Convert the processed word to lowercase.- If the resulting word is not empty, add it to the dictionary.But how to remove the punctuation? Maybe using the string's translate method, but that might be a bit involved. Alternatively, for each token, we can iterate from the start until we find a letter, then from the end until we find a letter, and take the substring in between.Wait, perhaps a better way is to use the string's `strip` method with a set of punctuation characters. But wait, the `strip` method removes all occurrences of the characters in the given set from both ends. So for example, if the token is "hello!", then token.strip(string.punctuation) would give "hello".Yes, that makes sense. So first, I'll import the string module, which has a string.punctuation that contains all punctuation marks.So the steps in code:1. Import necessary modules: string and re maybe, but perhaps not needed if I handle it with string methods.2. Initialize an empty dictionary to hold the word counts.3. Split the lyrics into tokens using split(), which by default splits on whitespace.4. For each token in the tokens: a. Strip all leading and trailing punctuation using token.strip(string.punctuation). b. Convert the stripped token to lowercase. c. If the resulting word is not an empty string, check if it's in the dictionary. If yes, increment the count. If not, add it with a count of 1.Wait, but what about apostrophes within words, like "don't"? The strip method with punctuation will not remove the apostrophe because it's in the middle. So "don't" would remain as "don't", which is correct.Wait, but in the problem statement, the example shows that "hello," becomes "hello". So the punctuation is stripped from the ends, but any punctuation in the middle is kept as part of the word.So that approach should work.Let me test this logic with the sample input.Sample input: "Hello, hello, how are you? I am fine, thank you!"Split into tokens: ["Hello,", "hello,", "how", "are", "you?", "I", "am", "fine,", "thank", "you!"]Processing each token:- "Hello," → strip punctuation → "Hello" → lowercase → "hello"- "hello," → same → "hello"- "how" → remains "how"- "are" → remains "are"- "you?" → "you"- "I" → "i"- "am" → "am"- "fine," → "fine"- "thank" → "thank"- "you!" → "you"So the words are: hello, hello, how, are, you, i, am, fine, thank, you.So the counts are:hello:2, how:1, are:1, you:2, i:1, am:1, fine:1, thank:1.Which matches the expected output.Great. So the approach should work.Now, let's think about the code.First, import string.Then, function definition:def extract_words(lyrics):Initialize the dictionary:word_counts = {}Then, split the lyrics into tokens:tokens = lyrics.split()Then, loop over each token:for token in tokens: stripped = token.strip(string.punctuation) word = stripped.lower() if word: if word in word_counts: word_counts[word] +=1 else: word_counts[word] =1Wait, but what about tokens that are entirely punctuation, like "!!"? After stripping, they become empty string, so we skip them.Yes.So that's the code.Wait, but what about apostrophes in the middle? Like "don't" would be kept as "don't", which is correct.What about hyphenated words, like "state-of-the-art"? The split would treat it as a single token, and stripping punctuation (if any) would leave it as "state-of-the-art", which is correct.So the code should handle that.Testing another example: lyrics = "Hi! How's it going? I'm fine." → tokens are ["Hi!", "How's", "it", "going?", "I'm", "fine."]Processing:"Hi!" → "Hi" → "hi""How's" → "How's" → "how's""it" → "it""going?" → "going""I'm" → "i'm""fine." → "fine"So the words are: hi, how's, it, going, i'm, fine.Each appears once.So the dictionary would have those keys with count 1.Another test case: lyrics = "Hello, my name's Alice. Alice's cat is cute." → tokens are ["Hello,", "my", "name's", "Alice.", "Alice's", "cat", "is", "cute."]Processing:"Hello," → "hello""my" → "my""name's" → "name's""Alice." → "alice""Alice's" → "alice's""cat" → "cat""is" → "is""cute." → "cute"So the counts are:hello:1, my:1, name's:1, alice:1, alice's:1, cat:1, is:1, cute:1.Which is correct.So the code should handle all these cases.Now, what about empty string? If lyrics is an empty string, the function returns an empty dict.What about multiple spaces between words? The split() method handles that, as it splits on any whitespace, regardless of how many.So the code should be correct.So putting it all together:Import string.Function:def extract_words(lyrics): word_counts = {} tokens = lyrics.split() for token in tokens: stripped = token.strip(string.punctuation) word = stripped.lower() if word: if word in word_counts: word_counts[word] += 1 else: word_counts[word] = 1 return word_countsWait, but what about other characters, like numbers or underscores? The problem says to extract words, but perhaps the definition of a word is any sequence of characters separated by whitespace, but with leading and trailing punctuation stripped. So the code as written treats any token as a word, as long as after stripping punctuation, it's non-empty.So that's correct.Testing the sample input:lyrics = "Hello, hello, how are you? I am fine, thank you!"The function returns {'hello': 2, 'how': 1, 'are': 1, 'you': 2, 'i': 1, 'am': 1, 'fine': 1, 'thank': 1}, which is correct.Another test case: lyrics = "A man, a plan, a canal: Panama!" → tokens are ["A", "man,", "a", "plan,", "a", "canal:", "Panama!"]Processing:"A" → "a""man," → "man""a" → "a""plan," → "plan""a" → "a""canal:" → "canal""Panama!" → "panama"So the counts are:a:3, man:1, plan:1, canal:1, panama:1.So the function returns that.Yes, that's correct.So the code should handle all cases correctly.

answer：First, I need to establish the budget constraint equation based on the costs of importing dragon fruits and mangosteens. The cost per kilogram for dragon fruits is 3, and for mangosteens, it's 5. With a total budget of 2,000, the equation will be 3x + 5y = 2000.Next, I'll solve this equation for y in terms of x. To isolate y, I'll subtract 3x from both sides, resulting in 5y = 2000 - 3x. Then, I'll divide both sides by 5 to get y = (2000 - 3x)/5.Moving on to the second part, the distributor estimates that the demand for dragon fruits will be twice that for mangosteens, which gives the relationship x = 2y. I'll substitute this into the budget equation. Replacing x with 2y in the equation 3x + 5y = 2000 gives 3(2y) + 5y = 2000. Simplifying this, I get 6y + 5y = 2000, which combines to 11y = 2000. Solving for y, I find y = 2000/11 ≈ 181.82 kilograms.Using the relationship x = 2y, I'll calculate x by doubling y: x = 2 * (2000/11) ≈ 363.64 kilograms.Finally, I'll verify the calculations by plugging these values back into the original budget equation to ensure they total 2,000.

answer：Okay, so I need to figure out the total energy required for a concert where I'm playing 12 pieces. Each piece has an energy requirement given by the polynomial E(n) = 2n² + 3n - 5, where n is the position of the piece in the concert order. Then, I also need to check if the average energy per piece is within 30 Steelant units.First, let me understand the problem. I have to calculate the sum of E(n) from n=1 to n=12. That means I need to compute E(1) + E(2) + ... + E(12). Since E(n) is a quadratic function, the sum can be calculated using the formula for the sum of squares, the sum of linear terms, and the sum of constants.Let me write down the expression for the total energy, which I'll call S.S = Σ (from n=1 to 12) [2n² + 3n - 5]I can split this sum into three separate sums:S = 2Σn² + 3Σn - 5Σ1Where each sum is from n=1 to 12.I remember the formulas for these sums:1. The sum of squares: Σn² from 1 to N is N(N + 1)(2N + 1)/62. The sum of the first N natural numbers: Σn from 1 to N is N(N + 1)/23. The sum of 1 N times is just N.So, let's compute each part step by step.First, compute Σn² from 1 to 12:Using the formula, N=12:Σn² = 12*13*25 / 6Wait, let me compute that step by step.12*13 = 156156*25 = 3900Then divide by 6: 3900 / 6 = 650So, Σn² = 650Next, compute Σn from 1 to 12:Σn = 12*13 / 2 = (156)/2 = 78Then, Σ1 from 1 to 12 is just 12.So now, plug these back into the expression for S:S = 2*650 + 3*78 - 5*12Let me compute each term:2*650 = 13003*78 = 2345*12 = 60So, S = 1300 + 234 - 60Compute 1300 + 234 first: that's 1534Then subtract 60: 1534 - 60 = 1474So, the total energy required is 1474 Steelant units.Wait, let me double-check my calculations to make sure I didn't make a mistake.First, Σn²: 12*13*25 /612*13 is 156, 156*25 is 3900, divided by 6 is 650. That seems correct.Σn: 12*13/2 is 78. Correct.Σ1: 12. Correct.Then, 2*650 is 1300, 3*78 is 234, 5*12 is 60.Adding 1300 + 234: 1300 + 200 is 1500, plus 34 is 1534. Then subtract 60: 1534 - 60 is 1474. Yes, that seems right.So, total energy is 1474.Now, for the second part, I need to find the average energy per piece. Since there are 12 pieces, the average is total energy divided by 12.Average = 1474 / 12Let me compute that.1474 divided by 12.12*120 = 14401474 - 1440 = 34So, 120 + 34/1234/12 is approximately 2.833...So, total average is approximately 122.833... Steelant units.Wait, but the question says whether the average does not exceed 30 Steelant units. 122.83 is way higher than 30. So, the concert program as planned does not meet the criterion.Wait, hold on, that seems like a huge average. Let me check if I did the total energy correctly.Wait, 1474 divided by 12 is indeed approximately 122.83. That's way above 30. Hmm. Maybe I made a mistake in calculating the total energy.Wait, let me re-examine the function E(n) = 2n² + 3n -5.So, for each piece n, the energy is 2n² + 3n -5.So, when n=1, E(1)=2(1)+3(1)-5=2+3-5=0.Wait, that's zero? That seems odd. Maybe the energy can't be negative, but at n=1, it's zero.Wait, let me compute E(1):2*(1)^2 + 3*(1) -5 = 2 + 3 -5 = 0.Hmm, okay. So, the first piece requires zero energy? That might be an oversight, but the function is given, so I have to go with it.Similarly, E(2)=2*(4) + 3*(2) -5=8+6-5=9.E(3)=2*9 + 9 -5=18+9-5=22.E(4)=2*16 +12 -5=32+12-5=39.E(5)=2*25 +15 -5=50+15-5=60.E(6)=2*36 +18 -5=72+18-5=85.E(7)=2*49 +21 -5=98+21-5=114.E(8)=2*64 +24 -5=128+24-5=147.E(9)=2*81 +27 -5=162+27-5=184.E(10)=2*100 +30 -5=200+30-5=225.E(11)=2*121 +33 -5=242+33-5=270.E(12)=2*144 +36 -5=288+36-5=319.Now, let me sum these up step by step.E(1)=0E(2)=9E(3)=22E(4)=39E(5)=60E(6)=85E(7)=114E(8)=147E(9)=184E(10)=225E(11)=270E(12)=319Now, let's add them one by one:Start with 0.Add 9: total 9Add 22: 31Add 39: 70Add 60: 130Add 85: 215Add 114: 329Add 147: 476Add 184: 660Add 225: 885Add 270: 1155Add 319: 1474So, the total is indeed 1474. So, my initial calculation was correct.Therefore, the average is 1474 /12 ≈122.83, which is way above 30. So, the concert program does not meet the criterion.But wait, maybe I misread the question. Let me check.The question says: "the average energy per piece does not exceed 30 Steelant units." So, 122.83 is way over, so it doesn't meet the criterion.Alternatively, perhaps I made a mistake in interpreting the function. Let me check the function again: E(n) = 2n² + 3n -5.Yes, that's correct. So, the first piece is zero, which is a bit strange, but mathematically, it's correct.Alternatively, maybe the function is supposed to be E(n) = 2n² + 3n -5, but perhaps the units are different? Or maybe the function is supposed to be in a different form? The problem statement says it's in Steelant units, so I think it's correct.Alternatively, perhaps the function is E(n) = 2n² + 3n -5, but maybe it's supposed to be divided by something? The problem doesn't say that, so I think it's just as given.Therefore, I think my calculations are correct. The total energy is 1474, average is approximately 122.83, which is way above 30. So, the concert program does not meet the criterion.Wait, but maybe I should present the average as an exact fraction instead of a decimal. Let me compute 1474 divided by 12.1474 ÷12.12*122=14641474-1464=10So, 122 and 10/12, which simplifies to 122 and 5/6, or approximately 122.833...So, yes, exactly 122 5/6.Therefore, the average is 122 5/6 Steelant units, which is much higher than 30.So, the answer to part 2 is no, the concert program does not meet the criterion.Wait, but just to make sure, maybe I should check the sum another way. I used the formula for the sum of squares and linear terms, but when I added them individually, I got the same result. So, that's consistent.Alternatively, maybe the function is supposed to be E(n) = 2n² + 3n -5, but perhaps it's per some other measure? The problem says "energy needed to play the nth piece," so I think it's correct.Alternatively, maybe the function is supposed to be E(n) = 2n² + 3n -5, but perhaps it's in a different unit? No, the problem says Steelant units, so I think it's correct.Therefore, I think my conclusion is correct.**Final Answer**1. The total energy required is boxed{1474} Steelant units.2. The average energy per piece is approximately 122.83 Steelant units, which exceeds 30. Therefore, the concert program does not meet the criterion. The final answer is boxed{1474} for the total energy and the average does not meet the criterion.Wait, the question asks for two separate answers. The first is the total energy, which is 1474, and the second is whether the average exceeds 30. So, perhaps the second answer is just a yes or no, but the problem says "determine whether your concert program, as currently planned, meets this criterion." So, the answer is no.But the instructions say to put the final answer within boxes. So, perhaps for part 1, it's 1474, and for part 2, it's no.But the initial instruction says "put your final answer within boxed{}." So, maybe I should present both answers in boxes.Alternatively, perhaps the second answer is just a statement, but since the user said "put your final answer within boxed{}", maybe I should box both.But the first answer is a number, the second is a yes/no. Hmm.Alternatively, perhaps the second answer can be expressed as a box with "No" or something.But the initial problem says "determine whether your concert program... meets this criterion." So, the answer is no, it does not meet the criterion.But in the instructions, it's said to put the final answer within boxes. So, perhaps I should write:1. boxed{1474}2. boxed{text{No}}But I'm not sure if the second answer should be boxed as text. Alternatively, perhaps just present both answers in separate boxes.Alternatively, maybe the second answer is just a numerical value, but since it's a yes/no, perhaps it's better to write it as text in a box.Alternatively, maybe the average is 122.83, which is greater than 30, so the answer is no.But perhaps the user expects both answers in boxes, so I'll do that.**Final Answer**1. The total energy required is boxed{1474} Steelant units.2. The concert program does not meet the criterion, so the answer is boxed{text{No}}.Alternatively, if the second answer is supposed to be numerical, perhaps it's the average, but the question asks whether it meets the criterion, so it's a yes/no.But since the user said to put the final answer within boxes, I think both answers should be boxed.So, final answers:1. boxed{1474}2. boxed{text{No}}

answer：Okay, so I have this problem where a political pundit is looking at how the length of prison sentences affects the recidivism rate. He's modeled this with a quadratic equation: R(x) = ax² + bx + c. My job is to find the coefficients a, b, and c, and then figure out the sentence length that minimizes the recidivism rate and what that minimum rate is.Alright, let's start with part 1. I need to find a, b, and c. I know that when x is 2, R(x) is 60; when x is 5, R(x) is 45; and when x is 10, R(x) is 30. So, I can set up three equations based on these points.First equation: When x = 2, R(2) = 60.So, plugging into the quadratic: a*(2)² + b*(2) + c = 60That simplifies to 4a + 2b + c = 60. Let me write that down as equation (1): 4a + 2b + c = 60.Second equation: When x = 5, R(5) = 45.So, a*(5)² + b*(5) + c = 45Which is 25a + 5b + c = 45. That's equation (2): 25a + 5b + c = 45.Third equation: When x = 10, R(10) = 30.So, a*(10)² + b*(10) + c = 30Which is 100a + 10b + c = 30. That's equation (3): 100a + 10b + c = 30.Now, I have three equations:1) 4a + 2b + c = 602) 25a + 5b + c = 453) 100a + 10b + c = 30I need to solve this system of equations to find a, b, and c. Let's see. Maybe I can subtract equation (1) from equation (2) to eliminate c.Equation (2) minus equation (1):(25a + 5b + c) - (4a + 2b + c) = 45 - 6025a - 4a + 5b - 2b + c - c = -1521a + 3b = -15Let me write that as equation (4): 21a + 3b = -15Similarly, subtract equation (2) from equation (3):(100a + 10b + c) - (25a + 5b + c) = 30 - 45100a - 25a + 10b - 5b + c - c = -1575a + 5b = -15That's equation (5): 75a + 5b = -15Now, I have two equations with two variables, a and b:Equation (4): 21a + 3b = -15Equation (5): 75a + 5b = -15I can try to solve these. Let's simplify equation (4) first. Let's divide equation (4) by 3:(21a)/3 + (3b)/3 = (-15)/37a + b = -5So, equation (4a): 7a + b = -5Similarly, equation (5): 75a + 5b = -15Let me divide equation (5) by 5:(75a)/5 + (5b)/5 = (-15)/515a + b = -3So, equation (5a): 15a + b = -3Now, I have:Equation (4a): 7a + b = -5Equation (5a): 15a + b = -3Now, subtract equation (4a) from equation (5a):(15a + b) - (7a + b) = -3 - (-5)15a - 7a + b - b = -3 + 58a = 2So, 8a = 2 => a = 2/8 => a = 1/4So, a is 1/4 or 0.25.Now, plug a back into equation (4a): 7a + b = -57*(1/4) + b = -57/4 + b = -5Convert 7/4 to decimal is 1.75, so 1.75 + b = -5Subtract 1.75 from both sides: b = -5 - 1.75 = -6.75So, b is -6.75 or -27/4.Now, with a and b known, we can find c from equation (1): 4a + 2b + c = 60Plugging in a = 1/4 and b = -27/4:4*(1/4) + 2*(-27/4) + c = 60Simplify:1 + (-54/4) + c = 60Convert -54/4 to decimal is -13.5So, 1 - 13.5 + c = 60Which is -12.5 + c = 60So, c = 60 + 12.5 = 72.5So, c is 72.5 or 145/2.So, coefficients are:a = 1/4b = -27/4c = 145/2Let me double-check these values with equation (3) to make sure.Equation (3): 100a + 10b + c = 30Plugging in:100*(1/4) + 10*(-27/4) + 145/2Calculate each term:100*(1/4) = 2510*(-27/4) = -270/4 = -67.5145/2 = 72.5So, 25 - 67.5 + 72.5 = ?25 - 67.5 is -42.5-42.5 + 72.5 is 30. Perfect, that matches equation (3). So, the coefficients are correct.So, part 1 is done. The quadratic equation is R(x) = (1/4)x² - (27/4)x + 145/2.Now, moving on to part 2: Determine the length of the prison sentence that minimizes the recidivism rate and calculate this minimum recidivism rate.Since R(x) is a quadratic function, and the coefficient of x² is positive (1/4), the parabola opens upwards, meaning the vertex is the minimum point.The x-coordinate of the vertex of a parabola given by ax² + bx + c is at x = -b/(2a).So, let's compute that.Given a = 1/4 and b = -27/4,x = -b/(2a) = -(-27/4) / (2*(1/4)) = (27/4) / (1/2) = (27/4) * (2/1) = 54/4 = 13.5So, the sentence length that minimizes recidivism is 13.5 years.Wait, that seems quite long. Let me double-check the calculation.x = -b/(2a)Given b is -27/4, so -b is 27/4.2a is 2*(1/4) = 1/2.So, (27/4) divided by (1/2) is (27/4)*(2/1) = 54/4 = 13.5. Yeah, that's correct.So, 13.5 years is where the minimum occurs.Now, to find the minimum recidivism rate, plug x = 13.5 into R(x).Let me compute R(13.5):R(13.5) = (1/4)*(13.5)² + (-27/4)*(13.5) + 145/2First, compute each term step by step.Compute (13.5)²: 13.5 * 13.513 * 13 = 16913 * 0.5 = 6.50.5 * 13 = 6.50.5 * 0.5 = 0.25So, 13.5 * 13.5 = (13 + 0.5)(13 + 0.5) = 13² + 2*13*0.5 + 0.5² = 169 + 13 + 0.25 = 182.25So, (1/4)*182.25 = 182.25 / 4 = 45.5625Next term: (-27/4)*(13.5)First, compute 27 * 13.527 * 10 = 27027 * 3.5 = 94.5So, 270 + 94.5 = 364.5So, 27 * 13.5 = 364.5Therefore, (-27/4)*13.5 = -364.5 / 4 = -91.125Third term: 145/2 = 72.5Now, add all three terms together:45.5625 - 91.125 + 72.5Compute 45.5625 - 91.125 first:45.5625 - 91.125 = -45.5625Then, -45.5625 + 72.5 = 26.9375So, R(13.5) = 26.9375%Wait, so the minimum recidivism rate is 26.9375%, which is 26.9375%.Hmm, that seems a bit low, but given the trend in the data points, it's decreasing as x increases, so maybe it's correct.Let me check the calculations again.First term: (1/4)*(13.5)^213.5 squared is 182.25, divided by 4 is 45.5625. Correct.Second term: (-27/4)*13.527*13.5 is 364.5, divided by 4 is 91.125, so negative is -91.125. Correct.Third term: 145/2 is 72.5. Correct.Adding them: 45.5625 - 91.125 + 72.545.5625 - 91.125 is -45.5625-45.5625 + 72.5 is 26.9375. Correct.So, 26.9375% is the minimum recidivism rate.Alternatively, 26.9375% can be expressed as a fraction. Let's see:0.9375 is 15/16, so 26.9375% is 26 and 15/16 percent, which is 431/16 percent.But as a decimal, 26.9375 is fine.So, summarizing part 2: The length of the prison sentence that minimizes the recidivism rate is 13.5 years, and the minimum recidivism rate is 26.9375%.Wait, but 13.5 years is 13 years and 6 months. That seems quite long, but perhaps in some jurisdictions, sentences can be that long. The model suggests that beyond 10 years, the recidivism rate continues to decrease, which might be counterintuitive, but according to the quadratic model, it's the case.Alternatively, maybe the model is only accurate within the given data range, but since the vertex is at 13.5, which is beyond the given x=10, the model predicts that recidivism rate would continue to decrease beyond 10 years, which might not be realistic, but mathematically, that's what the quadratic suggests.So, unless there's a mistake in the calculations, that's the result.Wait, let me check the quadratic equation again. Maybe I made a mistake in setting up the equations.Wait, when x=2, R=60; x=5, R=45; x=10, R=30.So, plotting these points, as x increases, R decreases, which is why the quadratic opens upwards, with the minimum at x=13.5. So, the model is that recidivism rate decreases as sentences get longer, up to 13.5 years, and then starts increasing again. But in reality, I don't know if longer sentences beyond a certain point would cause recidivism to increase, but according to this model, yes.But since the model is quadratic, it's symmetric around the vertex. So, if we had data beyond x=13.5, we would see recidivism rates increasing again.But with the given data points, the model is consistent.So, unless there's a calculation error, which I don't see, the answer is 13.5 years and 26.9375%.Wait, let me just check the vertex formula again. The x-coordinate is -b/(2a). Given a=1/4, b=-27/4.So, x = -(-27/4)/(2*(1/4)) = (27/4)/(1/2) = (27/4)*(2/1) = 54/4 = 13.5. Correct.So, yeah, that seems right.Therefore, the final answers are:1. a = 1/4, b = -27/4, c = 145/22. The sentence length that minimizes recidivism is 13.5 years, with a minimum rate of 26.9375%.**Final Answer**1. The coefficients are ( a = boxed{dfrac{1}{4}} ), ( b = boxed{-dfrac{27}{4}} ), and ( c = boxed{dfrac{145}{2}} ).2. The length of the prison sentence that minimizes the recidivism rate is ( boxed{13.5} ) years, and the minimum recidivism rate is ( boxed{26.9375%} ).

answer：Alright, so I've got this problem about modeling the trajectory of a Douglas A4 Skyhawk. It's given by the quadratic equation ( h(t) = -16t^2 + 120t + 500 ). There are two parts: first, finding the time when the plane reaches its maximum height and what that height is. Second, figuring out the time range when the plane is at least 300 feet above the ground for safety.Starting with the first part: finding the time when the maximum height occurs. I remember that for a quadratic function in the form ( h(t) = at^2 + bt + c ), the vertex gives the maximum or minimum point. Since the coefficient of ( t^2 ) is negative (-16), the parabola opens downward, meaning the vertex is the maximum point.The formula to find the time ( t ) at the vertex is ( t = -frac{b}{2a} ). In this equation, ( a = -16 ) and ( b = 120 ). Plugging those into the formula: ( t = -frac{120}{2*(-16)} ). Let me compute that.First, 2 times -16 is -32. So, ( t = -frac{120}{-32} ). Dividing 120 by 32, that's 3.75. Since both numerator and denominator are negative, the negatives cancel out, so ( t = 3.75 ) seconds. So, the maximum height occurs at 3.75 seconds.Now, to find the maximum height, I need to plug this value back into the original equation. So, ( h(3.75) = -16*(3.75)^2 + 120*(3.75) + 500 ). Let me compute each part step by step.First, ( (3.75)^2 ) is 14.0625. Then, multiplying by -16: -16 * 14.0625. Hmm, 16 * 14 is 224, and 16 * 0.0625 is 1, so total is 225. So, -16 * 14.0625 is -225.Next, 120 * 3.75. Let me compute that: 120 * 3 is 360, and 120 * 0.75 is 90, so total is 360 + 90 = 450.Adding all parts together: -225 + 450 + 500. Let's compute that. -225 + 450 is 225, and 225 + 500 is 725. So, the maximum height is 725 feet.Wait, let me double-check that calculation because sometimes I make arithmetic errors. So, ( h(3.75) = -16*(14.0625) + 120*(3.75) + 500 ). Calculating each term:-16 * 14.0625: 14.0625 * 16 is 225, so with the negative, it's -225.120 * 3.75: 3.75 is 3 and 3/4, so 120 * 3 is 360, 120 * 0.75 is 90, so total 450.Adding them up: -225 + 450 is 225, plus 500 is 725. Yeah, that seems correct.So, part 1 is done. The maximum height is 725 feet at 3.75 seconds.Moving on to part 2: ensuring the aircraft doesn't descend below 300 feet. So, we need to solve the inequality ( h(t) geq 300 ). That is, ( -16t^2 + 120t + 500 geq 300 ).First, let's subtract 300 from both sides to set the inequality to zero: ( -16t^2 + 120t + 500 - 300 geq 0 ). Simplifying that, 500 - 300 is 200, so the inequality becomes ( -16t^2 + 120t + 200 geq 0 ).Alternatively, I can write it as ( -16t^2 + 120t + 200 geq 0 ). To make it a bit easier, maybe factor out a common factor. Let's see, all coefficients are divisible by -8? Let me check: -16 divided by -8 is 2, 120 divided by -8 is -15, 200 divided by -8 is -25. Hmm, but that would make the inequality ( 2t^2 - 15t - 25 leq 0 ) because dividing by a negative number reverses the inequality sign. Alternatively, maybe factor out a negative sign.Alternatively, perhaps it's easier to multiply both sides by -1, which will reverse the inequality. So, ( 16t^2 - 120t - 200 leq 0 ). Hmm, but that might complicate things. Alternatively, just solve the quadratic equation ( -16t^2 + 120t + 200 = 0 ) to find the critical points.Let me write it as ( -16t^2 + 120t + 200 = 0 ). Multiply both sides by -1 to make it positive: ( 16t^2 - 120t - 200 = 0 ). Now, let's see if we can simplify this equation. All coefficients are divisible by 8: 16/8=2, 120/8=15, 200/8=25. So, dividing by 8, we get ( 2t^2 - 15t - 25 = 0 ).Now, let's solve this quadratic equation. Using the quadratic formula: ( t = frac{15 pm sqrt{(-15)^2 - 4*2*(-25)}}{2*2} ). Compute discriminant first: ( (-15)^2 = 225 ), ( 4*2*25 = 200 ). So, discriminant is 225 + 200 = 425. So, square root of 425. Let me see, 425 is 25*17, so sqrt(425) is 5*sqrt(17). Approximately, sqrt(17) is about 4.123, so 5*4.123 is about 20.615.So, the solutions are ( t = frac{15 pm 20.615}{4} ). Let's compute both roots.First, the positive root: ( 15 + 20.615 = 35.615 ), divided by 4 is approximately 8.90375 seconds.Second, the negative root: ( 15 - 20.615 = -5.615 ), divided by 4 is approximately -1.40375 seconds.Since time cannot be negative, we discard the negative root. So, the critical points are at approximately t ≈ -1.40375 and t ≈ 8.90375. But since time starts at t=0, the relevant interval is from t=0 to t≈8.90375.But wait, we have to remember that the original inequality was ( -16t^2 + 120t + 200 geq 0 ). The quadratic ( -16t^2 + 120t + 200 ) is a downward opening parabola, so it's positive between its roots. Since one root is negative and the other is positive, the interval where the quadratic is positive is from t ≈ -1.40375 to t ≈ 8.90375. But since t cannot be negative, the interval is from t=0 to t≈8.90375.But wait, let me think again. The original equation is ( h(t) = -16t^2 + 120t + 500 ). So, when we set ( h(t) geq 300 ), we get ( -16t^2 + 120t + 200 geq 0 ). The quadratic ( -16t^2 + 120t + 200 ) is positive between its roots. So, the solution is t between the two roots, but since one root is negative, the relevant interval is from t=0 to t≈8.90375.Wait, but let me verify this by testing a point in the interval. Let's pick t=0: h(0)=500, which is above 300. At t=8.90375, h(t)=300. So, the plane is above 300 feet from t=0 until t≈8.90375 seconds.But wait, let me check the calculation again because sometimes when we manipulate inequalities, especially multiplying by negatives, it can flip the inequality. Let me go back.Original inequality: ( -16t^2 + 120t + 500 geq 300 ). Subtract 300: ( -16t^2 + 120t + 200 geq 0 ). So, we have ( -16t^2 + 120t + 200 geq 0 ). The quadratic is negative outside the roots and positive between them. Since the parabola opens downward, it's positive between the two roots. So, the solution is t between the two roots, which are approximately -1.40375 and 8.90375. Since t cannot be negative, the solution is 0 ≤ t ≤ 8.90375.But let me compute the exact roots without approximating. The quadratic equation after simplifying was ( 2t^2 - 15t - 25 = 0 ). Using the quadratic formula:( t = frac{15 pm sqrt{225 + 200}}{4} = frac{15 pm sqrt{425}}{4} ).Simplify sqrt(425): 425 = 25*17, so sqrt(425) = 5*sqrt(17). So, the roots are ( t = frac{15 pm 5sqrt{17}}{4} ).So, the positive root is ( t = frac{15 + 5sqrt{17}}{4} ). Let me compute this exactly:First, compute sqrt(17): approximately 4.123105625617661.So, 5*sqrt(17) ≈ 5*4.123105625617661 ≈ 20.615528128088305.Then, 15 + 20.615528128088305 ≈ 35.615528128088305.Divide by 4: 35.615528128088305 / 4 ≈ 8.903882032022076 seconds.So, the exact positive root is ( frac{15 + 5sqrt{17}}{4} ) seconds, approximately 8.9039 seconds.Therefore, the plane is above 300 feet from t=0 until t≈8.9039 seconds.Wait, but let me think again. The quadratic ( -16t^2 + 120t + 200 geq 0 ) is positive between its roots, which are at t≈-1.40375 and t≈8.90375. So, the solution is t between these two values. But since t cannot be negative, the solution is t from 0 to approximately 8.90375 seconds.But wait, let me test t=9 seconds to see what h(t) is. Plugging t=9 into h(t):h(9) = -16*(81) + 120*9 + 500 = -1296 + 1080 + 500 = (-1296 + 1080) = -216 + 500 = 284. So, h(9)=284, which is below 300. So, at t=9, it's below 300. Therefore, the plane descends below 300 feet after approximately 8.90375 seconds.Similarly, at t=8.90375, h(t)=300.So, the range of time when the plane is at least 300 feet is from t=0 to t≈8.90375 seconds.But let me express the exact value. The positive root is ( t = frac{15 + 5sqrt{17}}{4} ). So, the exact time is ( t = frac{15 + 5sqrt{17}}{4} ) seconds.Alternatively, we can write it as ( t = frac{5(3 + sqrt{17})}{4} ), but perhaps it's better to leave it as ( frac{15 + 5sqrt{17}}{4} ).So, summarizing part 2: the aircraft maintains a height of at least 300 feet from t=0 until t= ( frac{15 + 5sqrt{17}}{4} ) seconds, which is approximately 8.90375 seconds.Wait, but let me check if I did everything correctly. Sometimes when dealing with quadratics and inequalities, especially after manipulation, it's easy to make a mistake.Original inequality: ( -16t^2 + 120t + 500 geq 300 ).Subtract 300: ( -16t^2 + 120t + 200 geq 0 ).This is a quadratic in standard form. Since the coefficient of ( t^2 ) is negative, the parabola opens downward. Therefore, the quadratic is positive between its two roots. The roots are at t≈-1.40375 and t≈8.90375. Since time cannot be negative, the interval where the quadratic is positive is from t=0 to t≈8.90375.Yes, that seems correct.Alternatively, another way to approach this is to find when h(t) = 300, which gives the times when the plane is exactly at 300 feet. Since the plane starts at h(0)=500, which is above 300, and since the parabola opens downward, the plane will be above 300 feet until it descends below 300 at t≈8.90375 seconds.So, the range of time is from t=0 to t= ( frac{15 + 5sqrt{17}}{4} ) seconds.To express this as an interval, it's [0, ( frac{15 + 5sqrt{17}}{4} )].But let me compute ( frac{15 + 5sqrt{17}}{4} ) exactly. Let's factor out 5: ( frac{5(3 + sqrt{17})}{4} ). So, that's another way to write it.Alternatively, we can rationalize or simplify further, but I think that's as simplified as it gets.So, to recap:1. The maximum height occurs at t=3.75 seconds, and the height is 725 feet.2. The plane is above 300 feet from t=0 to t= ( frac{15 + 5sqrt{17}}{4} ) seconds, which is approximately 8.90375 seconds.I think that's all. Let me just make sure I didn't make any calculation errors.For part 1:- Vertex at t= -b/(2a) = -120/(2*(-16)) = 120/32 = 3.75. Correct.- Plugging back in: h(3.75) = -16*(3.75)^2 + 120*(3.75) + 500.- 3.75 squared is 14.0625. 14.0625 * 16 = 225, so -225.- 120*3.75=450.- So, -225 + 450 = 225; 225 + 500=725. Correct.For part 2:- Set h(t)=300: -16t^2 + 120t + 500=300.- Subtract 300: -16t^2 + 120t + 200=0.- Multiply by -1: 16t^2 -120t -200=0.- Divide by 8: 2t^2 -15t -25=0.- Quadratic formula: t=(15±sqrt(225+200))/4=(15±sqrt(425))/4=(15±5sqrt(17))/4.- Positive root: (15+5sqrt(17))/4≈8.90375.Yes, all steps check out.

answer：Okay, so I have this problem about calculating the total power consumption of an MRI machine during a scan. The scan took 45 minutes, and the power consumption is modeled by the polynomial ( P(t) = 2t^2 - 3t + 5 ). There are two parts to the problem: first, calculating the total power consumption over the 45 minutes, and second, considering a more energy-efficient machine that reduces power consumption by 20%.Starting with part 1. The question mentions evaluating the polynomial at ( t = 0 ) and ( t = 45 ), and then finding the definite integral of ( P(t) ) from 0 to 45. Hmm, wait, evaluating at those points and then integrating? Let me think. I know that integrating the power function over time gives the total energy consumed, right? So, the total power consumption is actually the integral of ( P(t) ) from 0 to 45. But why does it mention evaluating at t=0 and t=45? Maybe that's just to get the power at the start and end times, but the integral is what gives the total consumption.Let me write down the integral. The total energy ( E ) is given by:[E = int_{0}^{45} P(t) , dt = int_{0}^{45} (2t^2 - 3t + 5) , dt]Alright, so I need to compute this integral. Let me recall how to integrate polynomials term by term.First, the integral of ( 2t^2 ) is ( frac{2}{3}t^3 ). Then, the integral of ( -3t ) is ( -frac{3}{2}t^2 ). The integral of 5 is ( 5t ). So putting it all together, the antiderivative ( F(t) ) is:[F(t) = frac{2}{3}t^3 - frac{3}{2}t^2 + 5t]Now, I need to evaluate this from 0 to 45. So, ( E = F(45) - F(0) ).Let me compute ( F(45) ):First term: ( frac{2}{3} times 45^3 ). Let me calculate 45 cubed. 45 squared is 2025, so 45 cubed is 45 * 2025. Let me compute that:45 * 2000 = 90,00045 * 25 = 1,125So, 90,000 + 1,125 = 91,125. So, 45^3 = 91,125.Then, ( frac{2}{3} times 91,125 ). Let me compute that:Divide 91,125 by 3: 91,125 / 3 = 30,375Then multiply by 2: 30,375 * 2 = 60,750So, the first term is 60,750.Second term: ( -frac{3}{2} times 45^2 ). We already know 45 squared is 2025.So, ( -frac{3}{2} times 2025 ). Let's compute that:2025 / 2 = 1012.51012.5 * 3 = 3037.5So, with the negative sign, it's -3037.5Third term: 5 * 45 = 225So, adding all three terms together:60,750 - 3,037.5 + 225Let me compute step by step:60,750 - 3,037.5 = 57,712.557,712.5 + 225 = 57,937.5So, ( F(45) = 57,937.5 ) kilowatt-minutes.Now, ( F(0) ) is simply plugging in 0 into the antiderivative:( frac{2}{3}(0)^3 - frac{3}{2}(0)^2 + 5(0) = 0 )So, the total energy ( E = 57,937.5 - 0 = 57,937.5 ) kilowatt-minutes.Wait, but power is in kilowatts, and integrating over minutes gives kilowatt-minutes. Is that the standard unit? Or should it be converted to kilowatt-hours? Since 1 hour is 60 minutes, 45 minutes is 0.75 hours. But the question just asks for total power consumption, so maybe it's okay to leave it in kilowatt-minutes. Let me check the question again."Calculate the total power consumption for the entire duration of the MRI scan..." It doesn't specify the unit, but the function is in kilowatts. So, integrating kilowatts over minutes gives kilowatt-minutes. Alternatively, if we wanted it in kilowatt-hours, we could divide by 60. But since the question doesn't specify, I think kilowatt-minutes is acceptable. Alternatively, maybe they expect kilowatt-hours. Hmm.Wait, let me think again. Power is in kilowatts, so energy is power multiplied by time. So, the unit would be kilowatt-minutes. But sometimes, energy is expressed in kilowatt-hours. So, perhaps the answer should be converted to kilowatt-hours by dividing by 60.Let me compute that: 57,937.5 / 60 = ?57,937.5 divided by 60:First, 57,937.5 / 60 = (57,937.5 ÷ 10) / 6 = 5,793.75 / 65,793.75 / 6: 6 * 965 = 5,790, so 965 with a remainder of 3.75.3.75 / 6 = 0.625So, total is 965.625 kilowatt-hours.But the question didn't specify, so maybe both are acceptable. But since the function is given in kilowatts and time is in minutes, the integral is in kilowatt-minutes. So, perhaps 57,937.5 kilowatt-minutes is the answer. Alternatively, if converted, 965.625 kilowatt-hours.Wait, but in the problem statement, it says "total power consumption". Hmm, power is in kilowatts, but power consumption over time is energy. So, maybe they just want the integral, which is energy, in kilowatt-minutes. Alternatively, they might expect it in kilowatt-hours. I think I should compute both, but probably kilowatt-minutes is the direct answer.But let me check the question again: "Calculate the total power consumption for the entire duration of the MRI scan by evaluating the polynomial at ( t = 0 ) and ( t = 45 ), and then find the definite integral of ( P(t) ) from ( t = 0 ) to ( t = 45 )."Wait, so they mention evaluating at t=0 and t=45, but then also integrating. So, perhaps they want both the power at start and end, and the total energy? Or maybe they just want the integral, which is the total energy. Hmm.But the question says "total power consumption", which is a bit ambiguous. If it's total energy consumed, then it's the integral. If it's average power, then it's the integral divided by time. But the wording is "total power consumption", which is a bit unclear. But in engineering terms, power consumption over time is energy. So, I think it's safe to go with the integral, which is 57,937.5 kilowatt-minutes or 965.625 kilowatt-hours.But since the function is given in kilowatts and time in minutes, maybe they expect the answer in kilowatt-minutes. Let me stick with that unless told otherwise.So, moving on. The first part is done. Now, part 2: the father is considering a more energy-efficient machine that reduces power consumption by 20%. So, the new power consumption is ( Q(t) = 0.8P(t) ). We need to express ( Q(t) ) as a polynomial and calculate the total power consumption over the same duration.Expressing ( Q(t) ) is straightforward. Since ( P(t) = 2t^2 - 3t + 5 ), multiplying by 0.8 gives:( Q(t) = 0.8(2t^2 - 3t + 5) )Let me compute that:0.8 * 2t^2 = 1.6t^20.8 * (-3t) = -2.4t0.8 * 5 = 4So, ( Q(t) = 1.6t^2 - 2.4t + 4 )Now, to find the total power consumption, we need to integrate ( Q(t) ) from 0 to 45.So, the integral is:[E' = int_{0}^{45} Q(t) , dt = int_{0}^{45} (1.6t^2 - 2.4t + 4) , dt]Again, integrating term by term.First term: integral of 1.6t^2 is ( frac{1.6}{3}t^3 ) which is ( frac{8}{15}t^3 ) or approximately 0.5333t^3.Second term: integral of -2.4t is ( -frac{2.4}{2}t^2 = -1.2t^2 )Third term: integral of 4 is 4t.So, the antiderivative ( F'(t) ) is:[F'(t) = frac{8}{15}t^3 - 1.2t^2 + 4t]Now, evaluate from 0 to 45.First, compute ( F'(45) ):First term: ( frac{8}{15} times 45^3 ). We already know 45^3 is 91,125.So, ( frac{8}{15} times 91,125 ). Let me compute that:Divide 91,125 by 15: 91,125 / 15 = 6,075Then multiply by 8: 6,075 * 8 = 48,600Second term: ( -1.2 times 45^2 ). 45 squared is 2025.So, ( -1.2 times 2025 = -2,430 )Third term: 4 * 45 = 180So, adding all three terms:48,600 - 2,430 + 180Compute step by step:48,600 - 2,430 = 46,17046,170 + 180 = 46,350So, ( F'(45) = 46,350 ) kilowatt-minutes.( F'(0) ) is 0, so the total energy ( E' = 46,350 - 0 = 46,350 ) kilowatt-minutes.Alternatively, in kilowatt-hours, that's 46,350 / 60 = 772.5 kilowatt-hours.But again, unless specified, I think kilowatt-minutes is acceptable.Wait, but let me think again. Since the original total was 57,937.5 kilowatt-minutes, and reducing by 20% should give us 80% of that, which is 0.8 * 57,937.5 = 46,350 kilowatt-minutes. So, that matches. So, that's a good check.So, to recap:1. Total power consumption with the original machine is 57,937.5 kilowatt-minutes.2. With the new machine, it's 46,350 kilowatt-minutes.Alternatively, in kilowatt-hours, that's approximately 965.625 and 772.5 respectively.But since the question didn't specify, I think the first answer is in kilowatt-minutes.Wait, but let me make sure. The question says "total power consumption". Power is in kilowatts, so total power consumption over time is energy, which is in kilowatt-minutes or kilowatt-hours. Since the scan took 45 minutes, which is less than an hour, maybe kilowatt-minutes is more precise. But sometimes, energy is expressed in kilowatt-hours regardless of the duration. Hmm.Alternatively, perhaps the question expects the answer in kilowatt-minutes because the time is in minutes. So, I think it's safe to present both answers, but since the integral is in kilowatt-minutes, that's the direct answer.Wait, but let me check the first part again. The question says: "Calculate the total power consumption for the entire duration of the MRI scan by evaluating the polynomial at ( t = 0 ) and ( t = 45 ), and then find the definite integral of ( P(t) ) from ( t = 0 ) to ( t = 45 )."So, they mention evaluating at t=0 and t=45, but then also integrating. So, perhaps they just want the integral, which is the total energy. So, the total power consumption over the duration is the integral, which is 57,937.5 kilowatt-minutes.Similarly, for the second part, the total power consumption is 46,350 kilowatt-minutes.Alternatively, if they want it in kilowatt-hours, we can convert it.But since the problem didn't specify, and the function is given in kilowatts with time in minutes, I think the answer is expected in kilowatt-minutes.So, to sum up:1. The total power consumption is 57,937.5 kilowatt-minutes.2. With the new machine, it's 46,350 kilowatt-minutes.Alternatively, if converted to kilowatt-hours, it's approximately 965.625 and 772.5 respectively.But since the question didn't specify, I think the first answer is in kilowatt-minutes.Wait, but let me think again. Maybe they just want the integral, which is 57,937.5, and then 46,350. So, I think that's the answer.But just to make sure, let me compute the integral again.For part 1:Integral of ( 2t^2 - 3t + 5 ) from 0 to 45.Antiderivative: ( frac{2}{3}t^3 - frac{3}{2}t^2 + 5t )At 45:( frac{2}{3}*(45)^3 = frac{2}{3}*91,125 = 60,750 )( -frac{3}{2}*(45)^2 = -frac{3}{2}*2025 = -3,037.5 )( 5*45 = 225 )Total: 60,750 - 3,037.5 + 225 = 57,937.5Yes, that's correct.For part 2:Integral of ( 1.6t^2 - 2.4t + 4 ) from 0 to 45.Antiderivative: ( frac{1.6}{3}t^3 - frac{2.4}{2}t^2 + 4t ) = ( frac{8}{15}t^3 - 1.2t^2 + 4t )At 45:( frac{8}{15}*91,125 = 48,600 )( -1.2*2025 = -2,430 )( 4*45 = 180 )Total: 48,600 - 2,430 + 180 = 46,350Yes, that's correct.So, I think I'm confident with these answers.