answer：First, I need to convert the base-12 number (1531_{12}) to its decimal (base-10) equivalent. To do this, I'll expand each digit by its place value in base-12.Starting from the rightmost digit:- The digit 1 is in the (12^0) place, which is (1 times 1 = 1).- The digit 3 is in the (12^1) place, which is (3 times 12 = 36).- The digit 5 is in the (12^2) place, which is (5 times 144 = 720).- The digit 1 is in the (12^3) place, which is (1 times 1728 = 1728).Adding all these values together gives the decimal equivalent:(1728 + 720 + 36 + 1 = 2485).Now, I'll divide this decimal number by 8 to find the remainder. Dividing 2485 by 8:- (8 times 310 = 2480).- Subtracting this from 2485 gives a remainder of 5.Therefore, the remainder when (1531_{12}) is divided by 8 is 5.

answer：First, I need to determine the total distance Qingqing covers in 7 minutes. Since she walks at a speed of 70 meters per minute, I can calculate her distance by multiplying her speed by the time: 70 meters/minute * 7 minutes = 490 meters.Next, I'll find out how much distance Linlin covers in the same time. The total distance between them initially is 3290 meters. Subtracting Qingqing's distance from the total gives Linlin's distance: 3290 meters - 490 meters = 2800 meters.Finally, to find Linlin's cycling speed, I'll divide the distance she covers by the time taken: 2800 meters / 7 minutes = 400 meters per minute.

answer：To solve the first expression, (12 - (-18) + (-7) - 15), I start by simplifying the subtraction of a negative number, which becomes addition. This transforms the expression into (12 + 18 + (-7) - 15). Next, I add the positive numbers: (12 + 18 = 30). Then, I subtract (7) from (30) to get (23), and finally subtract (15) to arrive at (8).For the second expression, ((-frac{1}{2}) times (-8) + (-6) div (-frac{1}{3})^{2}), I begin by calculating each term separately. Multiplying (-frac{1}{2}) by (-8) gives (4). For the division part, I first square (-frac{1}{3}) to get (frac{1}{9}), then divide (-6) by (frac{1}{9}), which results in (-54). Adding these two results together, (4 + (-54)), gives (-50).

answer：First, I identify the number of each type of bead in Sue's necklace.Sue has 7 purple beads.Next, since she has twice as many blue beads as purple beads, I calculate the number of blue beads by multiplying the number of purple beads by 2. This gives me 7 * 2 = 14 blue beads.Then, Sue has 11 more green beads than blue beads. To find the number of green beads, I add 11 to the number of blue beads: 14 + 11 = 25 green beads.Finally, to determine the total number of beads in the necklace, I add up all the beads: 7 purple + 14 blue + 25 green = 46 beads in total.

answer：Okay, so I have this problem here: The sum of the fourth and fifth powers of a certain non-integer number is known. Is it always possible to determine the sign of the original number? Hmm, interesting. Let me try to figure this out step by step.First, let me restate the problem to make sure I understand it. We have a number, let's call it x, which is not an integer. We know the sum of its fourth power and fifth power, which is x⁴ + x⁵. The question is asking whether, given this sum, we can always determine whether x is positive or negative.Alright, so I need to figure out if knowing x⁴ + x⁵ is enough information to tell if x is positive or negative. Let me think about the properties of even and odd powers.I know that for any real number x, the fourth power, x⁴, is always non-negative because any number raised to an even power is positive or zero. Similarly, the fifth power, x⁵, will have the same sign as x because it's an odd power. So, if x is positive, x⁵ is positive, and if x is negative, x⁵ is negative.So, the sum x⁴ + x⁵ will be the sum of a non-negative number and a number that can be positive or negative depending on x. Let me consider both cases: x positive and x negative.Case 1: x is positive.In this case, both x⁴ and x⁵ are positive. So, their sum is definitely positive. The question is, can this sum be equal to the sum when x is negative?Case 2: x is negative.Here, x⁴ is still positive, but x⁵ is negative. So, the sum x⁴ + x⁵ is the sum of a positive and a negative number. Depending on the magnitude, this could be positive or negative. Hmm, so it's possible that for some negative x, the sum x⁴ + x⁵ is positive, and for others, it might be negative.Wait, but the problem states that x is a non-integer. So, x is not an integer, but it could still be positive or negative. So, if we have x⁴ + x⁵, can we always tell if x was positive or negative?Let me try to analyze the function f(x) = x⁴ + x⁵. Maybe if I can show that this function is injective or something, or that it's monotonic, then knowing f(x) would allow us to determine x's sign.Let me compute the derivative of f(x) to see its behavior. The derivative f’(x) = 4x³ + 5x⁴. Hmm, let's factor that: x³(4 + 5x). So, critical points at x = 0 and x = -4/5.So, the function f(x) has critical points at x = 0 and x = -4/5. Let me analyze the intervals.For x > 0: Both x³ and (4 + 5x) are positive, so f’(x) is positive. So, f(x) is increasing for x > 0.For x between -4/5 and 0: x³ is negative, and (4 + 5x) is positive because 5x > -4 when x > -4/5. So, f’(x) is negative in this interval.For x < -4/5: x³ is negative, and (4 + 5x) is negative because 5x < -4. So, f’(x) is positive here because negative times negative is positive.So, the function f(x) is increasing on (-∞, -4/5), decreasing on (-4/5, 0), and increasing on (0, ∞). Hmm, interesting. So, it has a local maximum at x = -4/5 and a local minimum at x = 0.Let me compute the value of f(x) at x = -4/5.f(-4/5) = (-4/5)^4 + (-4/5)^5.Calculating that:(-4/5)^4 = (256)/(625) ≈ 0.4096(-4/5)^5 = (-1024)/(3125) ≈ -0.32768So, f(-4/5) ≈ 0.4096 - 0.32768 ≈ 0.08192.So, f(-4/5) is approximately 0.08192, which is positive.At x = 0, f(0) = 0 + 0 = 0.So, the function f(x) has a local maximum at x = -4/5 of approximately 0.08192, and then it decreases to 0 at x = 0, and then increases again for x > 0.So, for x > 0, f(x) is increasing from 0 to infinity. For x < -4/5, f(x) is increasing from negative infinity up to approximately 0.08192 at x = -4/5.Wait, so for x < -4/5, f(x) is increasing but starting from negative infinity? Let me check that.Wait, as x approaches negative infinity, x⁴ is positive and x⁵ is negative. So, x⁴ + x⁵ as x approaches negative infinity: x⁴ dominates because it's positive and growing faster than x⁵, which is negative. Wait, actually, x⁵ grows faster in magnitude than x⁴, but since it's negative, the sum x⁴ + x⁵ would approach negative infinity as x approaches negative infinity.Wait, that contradicts my earlier thought. Let me compute the limit as x approaches negative infinity of x⁴ + x⁵.We can factor out x⁴: x⁴(1 + x). As x approaches negative infinity, x⁴ approaches positive infinity, and (1 + x) approaches negative infinity. So, positive infinity times negative infinity is negative infinity. So, f(x) approaches negative infinity as x approaches negative infinity.So, f(x) is increasing on (-∞, -4/5), but it goes from negative infinity up to f(-4/5) ≈ 0.08192.Then, it decreases from x = -4/5 to x = 0, going from ~0.08192 down to 0.Then, for x > 0, it increases from 0 to infinity.So, the function f(x) = x⁴ + x⁵ is not injective over all real numbers because it's not one-to-one. For example, there might be two different x values, one positive and one negative, that give the same f(x). But wait, in this case, for x > 0, f(x) is increasing, so it's injective there. For x < -4/5, f(x) is increasing as well, but starting from negative infinity up to ~0.08192.Wait, so if we have a value of f(x) greater than ~0.08192, it must come from x > 0 because for x < -4/5, the maximum f(x) can reach is ~0.08192. So, if someone tells me that f(x) is, say, 1, then x must be positive because f(x) only reaches 1 for positive x.But if f(x) is between 0 and ~0.08192, then it could come from either a positive x or a negative x. Wait, but for positive x, f(x) starts at 0 and increases, so f(x) in (0, ∞) corresponds to x in (0, ∞). For negative x, f(x) is increasing from negative infinity up to ~0.08192. So, if f(x) is in (0, ~0.08192), it could be from a positive x or a negative x.Wait, but hold on, for x positive, f(x) is x⁴ + x⁵, which is positive. For x negative, f(x) can be positive or negative. Specifically, for x negative, f(x) is positive when x⁴ > |x⁵|, which is when |x| < 1, because x⁴ / |x⁵| = 1 / |x|. So, when |x| < 1, x⁴ > |x⁵|, so f(x) is positive. When |x| > 1, x⁵ dominates, so f(x) is negative.Wait, that's a good point. So, for x negative:- If |x| < 1, then x⁴ + x⁵ is positive because x⁴ > |x⁵|.- If |x| = 1, then x⁴ + x⁵ = 1 + (-1) = 0.- If |x| > 1, then x⁴ + x⁵ is negative because |x⁵| > x⁴.So, for x negative:- If x is between -1 and 0, f(x) is positive.- If x is less than -1, f(x) is negative.So, f(x) can take positive values for both positive x and negative x between -1 and 0. Similarly, f(x) can take negative values only for x < -1.So, if someone tells me that f(x) is positive, it could be either x positive or x negative between -1 and 0. If f(x) is negative, then x must be less than -1.But in the problem statement, x is a non-integer. So, x is not an integer, but it could still be positive or negative.Wait, but if f(x) is negative, then x must be less than -1, so x is negative. So, in that case, we can determine the sign.But if f(x) is positive, it could be x positive or x negative between -1 and 0. So, in that case, we can't determine the sign for sure.But the problem says "the sum of the fourth and fifth powers of a certain non-integer number is known." So, if the sum is known, can we always determine the sign?Wait, but if the sum is positive, we can't always determine the sign because it could be positive or negative (but between -1 and 0). If the sum is negative, then we can determine it's negative.But the question is whether it's "always possible" to determine the sign. So, if the sum is positive, sometimes we can't determine the sign because it could be positive or negative (but non-integer). So, does that mean it's not always possible?Wait, but let me think again. Maybe even when the sum is positive, we can still figure out the sign based on the value of the sum.Wait, for x positive, f(x) is increasing from 0 to infinity. For x negative between -1 and 0, f(x) is increasing from f(-1) = 1 + (-1) = 0 up to f(-4/5) ≈ 0.08192.So, for x negative between -1 and 0, f(x) is in (0, ~0.08192). For x positive, f(x) is in (0, ∞). So, if the sum is greater than ~0.08192, then x must be positive. If the sum is between 0 and ~0.08192, then x could be positive or negative (but negative in that case is between -1 and 0). If the sum is negative, x is less than -1.So, if the sum is greater than ~0.08192, we can say x is positive. If it's between 0 and ~0.08192, we can't be sure—it could be positive or negative. If it's negative, x is negative.But the problem is asking, is it always possible to determine the sign? So, in some cases, yes, but in others, no. So, the answer would be no, it's not always possible.Wait, but hold on. The problem says "a certain non-integer number." So, maybe the number is not only non-integer but also not equal to -4/5 or something? Hmm, not sure.Wait, but let's think about specific examples.Suppose someone tells me that x⁴ + x⁵ = 0.1. Then, 0.1 is between 0 and ~0.08192? Wait, 0.1 is actually greater than 0.08192. Wait, 0.08192 is approximately 0.082. So, 0.1 is greater than that. So, in that case, x must be positive because for negative x, the maximum f(x) is ~0.082, so if the sum is 0.1, x must be positive.Wait, so if the sum is greater than ~0.082, x must be positive. If it's less than or equal to ~0.082, it could be positive or negative.Wait, but 0.082 is f(-4/5). So, for x negative, the maximum f(x) is ~0.082. So, if the sum is greater than that, x must be positive.Therefore, if the sum is greater than ~0.082, we can conclude x is positive. If the sum is less than or equal to ~0.082, we can't be sure—it could be positive or negative.But the problem says "the sum of the fourth and fifth powers of a certain non-integer number is known." So, if the sum is known, depending on its value, we might or might not be able to determine the sign.Wait, but maybe I need to think about the function f(x) = x⁴ + x⁵ more carefully. Is it possible that for some positive x and some negative x (between -1 and 0), f(x) is equal?Yes, because f(x) is continuous and for positive x, it's increasing from 0 to infinity, and for negative x between -1 and 0, it's increasing from 0 to ~0.082. So, there must be some overlap where f(x) is the same for a positive x and a negative x.Therefore, given a sum S = x⁴ + x⁵, if S is greater than ~0.082, then x must be positive. If S is between 0 and ~0.082, then x could be positive or negative. If S is negative, x must be negative.So, in some cases, we can determine the sign, and in others, we can't. Therefore, it's not always possible to determine the sign of the original number.Wait, but the problem says "a certain non-integer number." So, if the number is non-integer, does that affect anything? Because whether it's integer or not, the function f(x) behaves the same way.Wait, unless the number is an integer, but it's given as non-integer, so maybe that's just to specify that x is not an integer, but it's still a real number.So, in conclusion, if the sum is greater than ~0.082, x is positive. If it's negative, x is negative. But if it's between 0 and ~0.082, we can't tell. Therefore, it's not always possible to determine the sign.But wait, the problem is asking if it's always possible. So, the answer would be no, because in some cases, we can't determine the sign.Wait, but hold on. Let me think again. Maybe I can find a way to determine the sign even when the sum is between 0 and ~0.082.Suppose I have S = x⁴ + x⁵, and S is in (0, ~0.082). Then, x could be positive or negative. But can I find another equation or something to figure out the sign?Wait, but we only know S. We don't have any other information. So, with just S, we can't determine if x is positive or negative in that interval.Therefore, the answer is no, it's not always possible to determine the sign of the original number because there exist sums S where x could be positive or negative.Wait, but let me test with specific numbers.Let me pick x = 0.5, which is positive and non-integer.f(0.5) = (0.5)^4 + (0.5)^5 = 0.0625 + 0.03125 = 0.09375.Now, let me find a negative x such that f(x) = 0.09375.So, solve x⁴ + x⁵ = 0.09375 for x negative.Let me try x = -0.5.f(-0.5) = (-0.5)^4 + (-0.5)^5 = 0.0625 - 0.03125 = 0.03125.That's less than 0.09375.Try x = -0.4.f(-0.4) = (0.4)^4 + (-0.4)^5 = 0.0256 - 0.01024 = 0.01536.Still less.Wait, maybe x = -0.3.f(-0.3) = (0.3)^4 + (-0.3)^5 = 0.0081 - 0.00243 = 0.00567.Hmm, getting smaller.Wait, but earlier I thought that for x negative between -1 and 0, f(x) increases from 0 to ~0.082. So, maybe x = -0.8.f(-0.8) = (0.8)^4 + (-0.8)^5 = 0.4096 - 0.32768 = 0.08192.Ah, that's the maximum.So, if I take x = -0.8, f(x) = ~0.08192.If I take x = -0.75.f(-0.75) = (0.75)^4 + (-0.75)^5.Compute (0.75)^4: 0.75^2 = 0.5625; squared again: ~0.3164.(-0.75)^5: - (0.75)^5. 0.75^2 = 0.5625, 0.75^3 = 0.421875, 0.75^4 = 0.31640625, 0.75^5 = 0.2373046875. So, negative of that is -0.2373046875.So, f(-0.75) = 0.31640625 - 0.2373046875 ≈ 0.0791.So, less than 0.08192.Wait, so at x = -0.8, f(x) is ~0.08192, which is the maximum for negative x.So, if I have S = 0.09375, which is greater than 0.08192, then x must be positive.But if S is, say, 0.08, which is less than 0.08192, then x could be positive or negative.Wait, let me check x positive: solve x⁴ + x⁵ = 0.08.Let me try x = 0.4.f(0.4) = 0.4^4 + 0.4^5 = 0.0256 + 0.01024 = 0.03584.Too low.x = 0.5: 0.09375.x = 0.45.0.45^4: 0.45^2 = 0.2025; squared: ~0.0410.0.45^5: 0.45^4 * 0.45 ≈ 0.0410 * 0.45 ≈ 0.01845.So, f(0.45) ≈ 0.0410 + 0.01845 ≈ 0.05945.Still less than 0.08.x = 0.475.Compute 0.475^4 and 0.475^5.0.475^2 = 0.2256.0.475^4 = (0.2256)^2 ≈ 0.0509.0.475^5 = 0.0509 * 0.475 ≈ 0.0242.So, f(0.475) ≈ 0.0509 + 0.0242 ≈ 0.0751.Still less than 0.08.x = 0.49.0.49^2 = 0.2401.0.49^4 = (0.2401)^2 ≈ 0.0576.0.49^5 = 0.0576 * 0.49 ≈ 0.0282.So, f(0.49) ≈ 0.0576 + 0.0282 ≈ 0.0858.That's greater than 0.08.So, somewhere between x = 0.475 and x = 0.49, f(x) crosses 0.08.Similarly, for negative x, f(x) can reach up to ~0.08192 at x = -0.8.So, if I have S = 0.08, which is just below the maximum of f(x) for negative x, then x could be positive or negative.Wait, but actually, for x positive, f(x) is increasing, so for S = 0.08, there is exactly one positive x that satisfies f(x) = 0.08. Similarly, for x negative, since f(x) is increasing from x = -∞ to x = -4/5, and then decreasing from x = -4/5 to x = 0, but wait, no, for x negative, f(x) is increasing from x = -∞ to x = -4/5, reaching a maximum at x = -4/5, and then decreasing towards x = 0.So, for S = 0.08, which is less than the maximum f(-4/5) ≈ 0.08192, there are two negative x's that satisfy f(x) = 0.08: one less than -4/5 and one between -4/5 and 0.Wait, no, hold on. For x < -4/5, f(x) is increasing from -∞ to ~0.08192. So, for S = 0.08, which is less than 0.08192, there is exactly one x < -4/5 such that f(x) = 0.08.Similarly, for x between -4/5 and 0, f(x) is decreasing from ~0.08192 to 0. So, for S = 0.08, which is less than 0.08192, there is exactly one x in (-4/5, 0) such that f(x) = 0.08.Therefore, for S = 0.08, there are three solutions: one positive x, one x < -4/5, and one x between -4/5 and 0.Wait, is that possible?Wait, no, because for x positive, f(x) is strictly increasing, so only one positive solution.For x negative, f(x) is increasing on (-∞, -4/5) and decreasing on (-4/5, 0). So, for S between 0 and ~0.08192, there are two negative solutions: one less than -4/5 and one between -4/5 and 0.Therefore, for S = 0.08, there are three real solutions: one positive, one less than -4/5, and one between -4/5 and 0.Wait, but that can't be, because f(x) is a quintic function, so it can have up to five real roots, but in this case, we're solving f(x) = S, which is a horizontal line. So, depending on S, the number of solutions varies.But in our case, for S positive, how many solutions are there?For S > ~0.08192, only one positive solution.For S = ~0.08192, one positive solution and one negative solution at x = -4/5.For 0 < S < ~0.08192, three solutions: one positive, one less than -4/5, and one between -4/5 and 0.For S = 0, two solutions: x = 0 and x = -1.For S negative, one solution: x < -1.Wait, so if S is known, and it's positive, then depending on its value, there could be one or three solutions. If S is greater than ~0.08192, only one positive solution. If S is between 0 and ~0.08192, three solutions: one positive, two negative.Therefore, if someone tells me that S is positive, I can't be sure whether x is positive or negative because there could be both positive and negative solutions.But if S is negative, then x must be negative.So, in conclusion, if the sum S is negative, we can determine x is negative. If S is positive, we can't determine the sign because x could be positive or negative. If S is zero, x could be 0 or -1, but x is non-integer, so x can't be -1, so x must be 0, but x is non-integer, so x can't be 0 either. Wait, hold on.Wait, the problem says "a certain non-integer number." So, x is non-integer. So, if S = 0, x can't be 0 or -1 because those are integers. So, S can't be 0 because x is non-integer. So, S can't be 0.Therefore, if S is known, and it's positive, x could be positive or negative (non-integer). If S is negative, x must be negative (non-integer). So, in the case of S negative, we can determine x is negative. In the case of S positive, we can't determine the sign.Therefore, it's not always possible to determine the sign of the original number because when S is positive, x could be positive or negative.So, the answer is no, it's not always possible.Wait, but hold on. Let me think again. If x is non-integer, does that affect the number of solutions?For example, if S is positive, could it be that for non-integer x, the solutions are unique in sign?Wait, no, because for S positive, as we saw, there are both positive and negative non-integer solutions. For example, S = 0.09375, which is f(0.5), and also f(-0.8) ≈ 0.08192, which is less than 0.09375, so actually, for S = 0.09375, x must be positive because the maximum for negative x is ~0.08192. So, in that case, x is positive.Wait, so if S is greater than ~0.08192, x must be positive. If S is between 0 and ~0.08192, x could be positive or negative. If S is negative, x must be negative.So, if someone tells me S, I can say:- If S > ~0.08192: x is positive.- If S < 0: x is negative.- If 0 < S < ~0.08192: x could be positive or negative.Therefore, in some cases, we can determine the sign, and in others, we can't.But the problem is asking, "Is it always possible to determine the sign of the original number?" So, since there are cases where we can't determine the sign, the answer is no.But wait, hold on. Let me think about whether the function f(x) is injective on the non-integer reals.If x is non-integer, does that make f(x) injective? Probably not, because even excluding integers, there can still be multiple x's mapping to the same S.Therefore, the answer is no, it's not always possible to determine the sign.Wait, but let me think about another approach. Maybe using the fact that x is non-integer, we can somehow infer the sign.Wait, but how? The function f(x) = x⁴ + x⁵ is even in some sense? No, it's not even because x⁵ is odd.Wait, maybe if I consider the ratio of x⁵ to x⁴, which is x. So, if I have S = x⁴ + x⁵, then S = x⁴(1 + x). So, 1 + x = S / x⁴.But that might not help directly.Alternatively, maybe I can write S = x⁴(1 + x), so 1 + x = S / x⁴. But without knowing x, that doesn't help.Alternatively, if I take the ratio of S to x⁴, which is 1 + x, but again, without knowing x, it's not helpful.Wait, maybe if I take logarithms or something, but that might complicate.Alternatively, think about the function f(x) = x⁴ + x⁵ and its behavior.As we saw, for positive x, f(x) is strictly increasing, so injective. For negative x, it's increasing from -∞ to x = -4/5, then decreasing from x = -4/5 to 0.So, for S > ~0.08192, only positive x. For S between 0 and ~0.08192, both positive and negative x. For S < 0, only negative x.Therefore, if we know S, we can sometimes determine the sign, but not always.Therefore, the answer is no, it's not always possible.But wait, the problem says "a certain non-integer number." So, maybe "certain" implies that it's given that such a number exists, but we don't know its sign. So, given S, can we determine the sign?But from our previous analysis, it's not always possible.Therefore, the answer is no.But wait, let me think again. Maybe I'm overcomplicating.Suppose I have S = x⁴ + x⁵.If x is positive, S is positive.If x is negative, S can be positive or negative.But if x is negative and |x| > 1, S is negative.If x is negative and |x| < 1, S is positive.So, if S is negative, x must be negative.If S is positive, x could be positive or negative (with |x| < 1).Therefore, if S is negative, we know x is negative.If S is positive, we can't be sure.But the problem says "the sum of the fourth and fifth powers of a certain non-integer number is known." So, if S is known, can we determine the sign?If S is negative, yes, x is negative.If S is positive, we can't determine.Therefore, it's not always possible.Hence, the answer is no.**Final Answer**boxed{No}

answer：Okay, so I have this problem about a store selling football commemorative books during the FIFA World Cup in Russia. Let me try to understand what it's asking and figure out how to solve it step by step.First, the problem is divided into two parts. Part (1) asks for the function relationship between the daily sales volume y and the selling price x, along with the range of x. Part (2) is about finding the optimal selling price to maximize the daily profit w, and also determining that maximum profit.Let me start with part (1). The problem gives some initial information: the cost price of each book is 40 yuan. The selling price is set not less than 44 yuan, and the profit margin cannot exceed 30%. So, the selling price x has a minimum of 44 yuan. What about the maximum?Profit margin not exceeding 30% means that the selling price cannot be more than 30% higher than the cost price. Let me calculate that. The cost price is 40 yuan, so 30% of 40 is 0.3 * 40 = 12 yuan. Therefore, the maximum selling price is 40 + 12 = 52 yuan. So, the selling price x must satisfy 44 ≤ x ≤ 52 yuan.Next, the sales information: when the selling price is set at 44 yuan, 300 books are sold per day. For every increase of 1 yuan in the selling price, the daily sales decrease by 10 books. So, if the price goes up by 1 yuan, sales go down by 10 books. Therefore, the relationship between y and x is linear.Let me denote x as the selling price. The base price is 44 yuan, and for each yuan above 44, the sales decrease by 10. So, the number of yuan above 44 is (x - 44). Therefore, the decrease in sales is 10*(x - 44). So, the daily sales y can be expressed as:y = 300 - 10*(x - 44)Simplify that:y = 300 - 10x + 440Wait, hold on. 300 - 10*(x - 44) is 300 - 10x + 440? Wait, that would be 300 + 440 - 10x, which is 740 - 10x. Hmm, that seems high. Wait, let me check again.Wait, no. The decrease in sales is 10*(x - 44). So, if x is 44, then y is 300. If x is 45, then y is 300 - 10*(1) = 290. If x is 46, y is 280, and so on. So, the formula should be:y = 300 - 10*(x - 44)Which is y = 300 - 10x + 440, which is y = 740 - 10x.Wait, but when x is 44, y is 740 - 10*44 = 740 - 440 = 300. That's correct. When x is 45, y is 740 - 450 = 290. Correct. So, the function is y = 740 - 10x.But hold on, is that right? Because 300 - 10*(x - 44) is 300 -10x + 440, which is 740 -10x. So, yes, that's correct.But let me think again. If x is 44, y is 300. If x increases by 1, y decreases by 10. So, the slope is -10. So, the equation is linear with slope -10, passing through (44, 300). So, point-slope form is y - 300 = -10(x - 44). So, y = -10x + 440 + 300 = -10x + 740. So, yes, that's correct.So, the function is y = -10x + 740, or y = 740 -10x.Now, the range of x is given by the constraints: the selling price must be at least 44 yuan and at most 52 yuan because the profit margin cannot exceed 30%. So, x ∈ [44, 52].Therefore, for part (1), the function is y = 740 - 10x, with x ranging from 44 to 52.Moving on to part (2). We need to find the selling price x that maximizes the daily profit w. To do this, I need to express the profit w in terms of x and then find its maximum.Profit is calculated as (selling price - cost price) * number of books sold. So, w = (x - 40) * y.We already have y in terms of x, so we can substitute that in.So, w = (x - 40)*(740 - 10x)Let me expand this expression:w = (x - 40)*(740 - 10x)Multiply term by term:First, x*740 = 740xx*(-10x) = -10x²-40*740 = -29,600-40*(-10x) = 400xSo, combining all terms:w = 740x -10x² -29,600 + 400xCombine like terms:740x + 400x = 1,140xSo, w = -10x² + 1,140x -29,600So, the profit function is a quadratic function in terms of x, and it's a downward opening parabola because the coefficient of x² is negative (-10). Therefore, the maximum occurs at the vertex.The vertex of a parabola given by ax² + bx + c is at x = -b/(2a). So, in this case, a = -10, b = 1,140.So, x = -1,140 / (2*(-10)) = -1,140 / (-20) = 57.Wait, but hold on. The vertex is at x = 57, but our x is constrained between 44 and 52. So, 57 is outside of our domain. Therefore, the maximum profit must occur at one of the endpoints of the interval.So, we need to evaluate the profit function at x = 44 and x = 52, and see which one gives a higher profit.But wait, let me double-check my calculation for the vertex. Maybe I made a mistake.Given w = -10x² + 1,140x -29,600So, a = -10, b = 1,140x = -b/(2a) = -1,140 / (2*(-10)) = -1,140 / (-20) = 57.Yes, that's correct. So, the vertex is at x = 57, which is beyond our maximum allowed x of 52. Therefore, the maximum profit within the domain [44, 52] will be at x = 52.But let me verify this by calculating the profit at x = 52 and x = 44, and maybe at some other points in between to see if perhaps the profit is higher somewhere else.Wait, but since the parabola is opening downward, and the vertex is at x = 57, which is beyond our maximum x of 52, the function is increasing on the interval [44, 52]. Therefore, the maximum profit occurs at x = 52.But let me compute the profit at x = 52 and x = 44 to confirm.First, at x = 52:y = 740 -10*52 = 740 - 520 = 220 books.Profit w = (52 - 40)*220 = 12*220 = 2,640 yuan.At x = 44:y = 740 -10*44 = 740 - 440 = 300 books.Profit w = (44 - 40)*300 = 4*300 = 1,200 yuan.So, clearly, at x = 52, the profit is higher than at x = 44.But wait, let me check another point, say x = 50.y = 740 -10*50 = 740 - 500 = 240 books.Profit w = (50 - 40)*240 = 10*240 = 2,400 yuan.Which is less than 2,640. So, indeed, as x increases, the profit increases until x = 52.Wait, but let me check x = 51.y = 740 -10*51 = 740 - 510 = 230 books.Profit w = (51 - 40)*230 = 11*230 = 2,530 yuan.Which is still less than 2,640.Similarly, x = 53 is beyond our maximum, but let's see:y = 740 -10*53 = 740 - 530 = 210 books.Profit w = (53 - 40)*210 = 13*210 = 2,730 yuan.But x = 53 is beyond our allowed maximum of 52. So, we can't consider that.Therefore, within the allowed range, the maximum profit is at x = 52, which gives 2,640 yuan.But wait, let me think again. The profit function is quadratic, and since the vertex is at x = 57, which is beyond our domain, the function is increasing on [44, 52]. So, the maximum is indeed at x = 52.But let me check if the profit function is indeed increasing throughout the interval.Take the derivative of w with respect to x:dw/dx = -20x + 1,140Set derivative equal to zero for critical points:-20x + 1,140 = 020x = 1,140x = 57So, the critical point is at x = 57, which is outside our domain. Therefore, on the interval [44, 52], the function is increasing because the derivative at x = 44 is:dw/dx = -20*44 + 1,140 = -880 + 1,140 = 260 > 0So, the function is increasing throughout the interval. Therefore, maximum at x = 52.Therefore, the optimal selling price is 52 yuan, and the maximum profit is 2,640 yuan.But let me just make sure I didn't make any calculation errors.Calculating y at x = 52:y = 740 -10*52 = 740 - 520 = 220.Profit w = (52 - 40)*220 = 12*220 = 2,640.Yes, that's correct.At x = 52, selling price is 52 yuan, which is 30% above cost (40*1.3 = 52), so that's within the profit margin constraint.So, all constraints are satisfied.Therefore, the answer for part (2) is that the selling price should be set at 52 yuan, and the maximum profit is 2,640 yuan.Wait, but just to be thorough, let me check the profit at x = 52 and x = 51, 50, etc., to see the trend.At x = 52: 220 books, profit 2,640.x = 51: 230 books, profit 2,530.x = 50: 240 books, profit 2,400.x = 49: 250 books, profit 2,250.x = 48: 260 books, profit 2,080.x = 47: 270 books, profit 1,920.x = 46: 280 books, profit 1,760.x = 45: 290 books, profit 1,600.x = 44: 300 books, profit 1,200.So, as x increases from 44 to 52, the profit increases each time, confirming that the maximum is at x = 52.Therefore, I'm confident that the optimal price is 52 yuan, with a maximum profit of 2,640 yuan.**Final Answer**(1) The function relationship is boxed{y = 740 - 10x} with the range of ( x ) being boxed{[44, 52]}.(2) The selling price should be set at boxed{52} yuan to maximize the daily profit, and the maximum profit is boxed{2640} yuan.