answer：First, I need to determine the work rates of Person A and Person B. Person A can complete the project in 10 days, so their work rate is ( frac{1}{10} ) of the project per day. Similarly, Person B can complete the project in 15 days, so their work rate is ( frac{1}{15} ) of the project per day.Next, I'll calculate the amount of work Person A completes in the first 3 days. Since Person A's work rate is ( frac{1}{10} ), in 3 days they complete ( frac{3}{10} ) of the project.After Person A stops working, Person B takes over and needs ( x ) additional days to complete the remaining work. Person B's work rate is ( frac{1}{15} ), so in ( x ) days, they complete ( frac{x}{15} ) of the project.To find the total work completed, I'll add the work done by Person A and Person B:[frac{3}{10} + frac{x}{15} = 1]This equation represents the total work done, which equals 1 complete project.

answer：Okay, so I have this equation: (x - 4)(3x + 11) = x² - 19x + 72. I need to find the two solutions, p and q, and then compute (p + 4)(q + 4). Hmm, let me think about how to approach this.First, I remember that to solve an equation like this, I should probably expand the left side and then bring all terms to one side to form a quadratic equation. Once I have a quadratic equation, I can use the quadratic formula or factorization to find the roots, which are p and q. After that, I can plug those roots into the expression (p + 4)(q + 4) and simplify it.Alright, let's start by expanding the left side: (x - 4)(3x + 11). I'll use the distributive property, also known as the FOIL method for binomials. First, multiply the First terms: x * 3x = 3x².Next, the Outer terms: x * 11 = 11x.Then, the Inner terms: -4 * 3x = -12x.Lastly, the Last terms: -4 * 11 = -44.Now, combining all these terms together: 3x² + 11x - 12x - 44. Let me simplify that by combining like terms. The 11x and -12x can be combined: 11x - 12x = -x. So, the left side simplifies to 3x² - x - 44.So now, the equation is 3x² - x - 44 = x² - 19x + 72. To solve for x, I need to bring all terms to one side. Let me subtract the right side from both sides to get everything on the left.So, subtracting x² - 19x + 72 from both sides:3x² - x - 44 - (x² - 19x + 72) = 0.Let me distribute the negative sign inside the parentheses:3x² - x - 44 - x² + 19x - 72 = 0.Now, combine like terms:3x² - x² = 2x².- x + 19x = 18x.-44 - 72 = -116.So, the equation simplifies to 2x² + 18x - 116 = 0.Hmm, this is a quadratic equation in the standard form ax² + bx + c = 0, where a = 2, b = 18, and c = -116.I can try to simplify this equation by dividing all terms by 2 to make the numbers smaller and easier to work with. Let's do that:2x² ÷ 2 = x².18x ÷ 2 = 9x.-116 ÷ 2 = -58.So, the simplified equation is x² + 9x - 58 = 0.Now, I need to find the solutions to this quadratic equation. Since it doesn't look easily factorable, I think I'll use the quadratic formula. The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a).In this case, a = 1, b = 9, and c = -58. Plugging these into the formula:x = [-9 ± √(9² - 4*1*(-58))] / (2*1).First, compute the discriminant, which is the part under the square root: 9² - 4*1*(-58).Calculating that:9² = 81.-4*1*(-58) = +232.So, the discriminant is 81 + 232 = 313.Therefore, the solutions are:x = [-9 ± √313] / 2.So, the two solutions are:p = [-9 + √313] / 2,q = [-9 - √313] / 2.Alright, now I need to compute (p + 4)(q + 4). Let me write that out:(p + 4)(q + 4) = ([-9 + √313]/2 + 4) * ([-9 - √313]/2 + 4).Hmm, let me simplify each term inside the parentheses first.Starting with p + 4:p + 4 = ([-9 + √313]/2) + 4.To add these, I can write 4 as 8/2 so that both terms have the same denominator.So, 4 = 8/2.Thus, p + 4 = ([-9 + √313]/2) + (8/2) = [(-9 + √313) + 8]/2.Simplify the numerator:-9 + 8 = -1.So, p + 4 = (-1 + √313)/2.Similarly, q + 4:q + 4 = ([-9 - √313]/2) + 4.Again, write 4 as 8/2:q + 4 = ([-9 - √313]/2) + (8/2) = [(-9 - √313) + 8]/2.Simplify the numerator:-9 + 8 = -1.So, q + 4 = (-1 - √313)/2.Now, the expression becomes:(p + 4)(q + 4) = [(-1 + √313)/2] * [(-1 - √313)/2].This looks like the product of two binomials in the form (a + b)(a - b), which is a difference of squares: a² - b².In this case, a = -1 and b = √313.So, applying the difference of squares formula:[(-1)² - (√313)²] / (2 * 2).Compute each part:(-1)² = 1.(√313)² = 313.So, the numerator becomes 1 - 313 = -312.The denominator is 4.Therefore, (p + 4)(q + 4) = -312 / 4 = -78.Wait, let me check my steps again to make sure I didn't make a mistake.First, expanding (x - 4)(3x + 11) gives 3x² - x - 44. That seems correct.Then, subtracting x² - 19x + 72 from both sides gives 2x² + 18x - 116 = 0. Dividing by 2 gives x² + 9x - 58 = 0. That all looks good.Using the quadratic formula: x = [-9 ± √(81 + 232)] / 2 = [-9 ± √313]/2. Correct.Then, adding 4 to each root:p + 4 = (-9 + √313)/2 + 4 = (-1 + √313)/2,q + 4 = (-9 - √313)/2 + 4 = (-1 - √313)/2.Multiplying these together: [(-1 + √313)/2] * [(-1 - √313)/2] = [(-1)^2 - (√313)^2]/4 = (1 - 313)/4 = (-312)/4 = -78.Yes, that seems consistent. So, the final answer is -78.I think that's it. I don't see any mistakes in my calculations, so I'm confident that (p + 4)(q + 4) equals -78.

answer：Alright, so I've got this problem here about a factory testing a new device against an old one to see if a certain indicator has improved. They've given me data for 10 products from each device, and I need to do two things: first, calculate the sample means and variances for both devices, and second, determine if the new device's mean has significantly improved based on a specific criterion.Okay, let's start with part (1). I need to find the sample means, which are denoted as (overline{x}) for the old device and (overline{y}) for the new device. Then, I also need to find the sample variances, (s_1^2) for the old device and (s_2^2) for the new device.First, I'll tackle the means. For the old device, the data points are: 9.8, 10.3, 10.0, 10.2, 9.9, 9.8, 10.0, 10.1, 10.2, 9.7. To find the mean, I need to add all these numbers together and then divide by 10, since there are 10 products.Let me add them up:9.8 + 10.3 = 20.120.1 + 10.0 = 30.130.1 + 10.2 = 40.340.3 + 9.9 = 50.250.2 + 9.8 = 60.060.0 + 10.0 = 70.070.0 + 10.1 = 80.180.1 + 10.2 = 90.390.3 + 9.7 = 100.0Wait, that's interesting. The total adds up to exactly 100.0. So, the mean (overline{x}) is 100.0 divided by 10, which is 10.0.Okay, that was straightforward. Now, for the new device, the data points are: 10.1, 10.4, 10.1, 10.0, 10.1, 10.3, 10.6, 10.5, 10.4, 10.5. Let's add these up.10.1 + 10.4 = 20.520.5 + 10.1 = 30.630.6 + 10.0 = 40.640.6 + 10.1 = 50.750.7 + 10.3 = 61.061.0 + 10.6 = 71.671.6 + 10.5 = 82.182.1 + 10.4 = 92.592.5 + 10.5 = 103.0So, the total is 103.0. Dividing by 10 gives the mean (overline{y}) as 10.3.Alright, so (overline{x} = 10.0) and (overline{y} = 10.3). That seems clear.Now, moving on to the variances. Variance is a measure of how spread out the numbers are. The formula for sample variance is:[s^2 = frac{1}{n-1} sum_{i=1}^{n} (x_i - overline{x})^2]But wait, in the problem statement, they just say "sample variances," and sometimes people use (n) instead of (n-1) in the denominator. I need to check if they specify which one to use. Looking back, the problem says "sample variances are denoted as (s_1^2) and (s_2^2)." It doesn't specify whether to use (n) or (n-1). Hmm.Well, in statistics, when we calculate sample variance, it's usually with (n-1) to get an unbiased estimate. But sometimes, especially in contexts like quality control, they might use (n). I need to see if the problem gives any clue. Looking at the criterion for significant improvement, it's:[overline{y} - overline{x} geqslant 2sqrt{frac{s_1^2 + s_2^2}{10}}]So, they're using (s_1^2) and (s_2^2) in the denominator with 10, which is the sample size. That suggests that they're using the population variance formula, which divides by (n), not (n-1). So, I think I should use (n) in the denominator for the variances here.Alright, so for the old device, (s_1^2), I need to calculate the squared differences from the mean, sum them up, and divide by 10.Let's list the data points for the old device again: 9.8, 10.3, 10.0, 10.2, 9.9, 9.8, 10.0, 10.1, 10.2, 9.7.Mean is 10.0, so let's compute each ((x_i - overline{x})^2):1. ( (9.8 - 10.0)^2 = (-0.2)^2 = 0.04 )2. ( (10.3 - 10.0)^2 = (0.3)^2 = 0.09 )3. ( (10.0 - 10.0)^2 = 0^2 = 0 )4. ( (10.2 - 10.0)^2 = (0.2)^2 = 0.04 )5. ( (9.9 - 10.0)^2 = (-0.1)^2 = 0.01 )6. ( (9.8 - 10.0)^2 = (-0.2)^2 = 0.04 )7. ( (10.0 - 10.0)^2 = 0^2 = 0 )8. ( (10.1 - 10.0)^2 = (0.1)^2 = 0.01 )9. ( (10.2 - 10.0)^2 = (0.2)^2 = 0.04 )10. ( (9.7 - 10.0)^2 = (-0.3)^2 = 0.09 )Now, let's sum these squared differences:0.04 + 0.09 = 0.130.13 + 0 = 0.130.13 + 0.04 = 0.170.17 + 0.01 = 0.180.18 + 0.04 = 0.220.22 + 0 = 0.220.22 + 0.01 = 0.230.23 + 0.04 = 0.270.27 + 0.09 = 0.36So, the sum of squared differences is 0.36. Now, divide by 10 to get (s_1^2):[s_1^2 = frac{0.36}{10} = 0.036]Okay, that's (s_1^2 = 0.036).Now, for the new device, (s_2^2). The data points are: 10.1, 10.4, 10.1, 10.0, 10.1, 10.3, 10.6, 10.5, 10.4, 10.5.Mean is 10.3, so let's compute each ((y_i - overline{y})^2):1. ( (10.1 - 10.3)^2 = (-0.2)^2 = 0.04 )2. ( (10.4 - 10.3)^2 = (0.1)^2 = 0.01 )3. ( (10.1 - 10.3)^2 = (-0.2)^2 = 0.04 )4. ( (10.0 - 10.3)^2 = (-0.3)^2 = 0.09 )5. ( (10.1 - 10.3)^2 = (-0.2)^2 = 0.04 )6. ( (10.3 - 10.3)^2 = 0^2 = 0 )7. ( (10.6 - 10.3)^2 = (0.3)^2 = 0.09 )8. ( (10.5 - 10.3)^2 = (0.2)^2 = 0.04 )9. ( (10.4 - 10.3)^2 = (0.1)^2 = 0.01 )10. ( (10.5 - 10.3)^2 = (0.2)^2 = 0.04 )Now, let's sum these squared differences:0.04 + 0.01 = 0.050.05 + 0.04 = 0.090.09 + 0.09 = 0.180.18 + 0.04 = 0.220.22 + 0 = 0.220.22 + 0.09 = 0.310.31 + 0.04 = 0.350.35 + 0.01 = 0.360.36 + 0.04 = 0.40So, the sum of squared differences is 0.40. Now, divide by 10 to get (s_2^2):[s_2^2 = frac{0.40}{10} = 0.04]Alright, so (s_2^2 = 0.04).Wait a second, let me double-check my calculations for the new device. The sum of squared differences was 0.40? Let me add them again:1. 0.042. 0.01 → total 0.053. 0.04 → total 0.094. 0.09 → total 0.185. 0.04 → total 0.226. 0 → total 0.227. 0.09 → total 0.318. 0.04 → total 0.359. 0.01 → total 0.3610. 0.04 → total 0.40Yes, that's correct. So, (s_2^2 = 0.04).Okay, so part (1) is done. I have:[overline{x} = 10.0, quad overline{y} = 10.3, quad s_1^2 = 0.036, quad s_2^2 = 0.04]Now, moving on to part (2). I need to determine whether the mean of the indicator for the new device has significantly improved compared to the old device. The criterion given is:[overline{y} - overline{x} geqslant 2sqrt{frac{s_1^2 + s_2^2}{10}}]If this inequality holds, then the improvement is considered significant; otherwise, it's not.First, let's compute (overline{y} - overline{x}):[overline{y} - overline{x} = 10.3 - 10.0 = 0.3]Next, let's compute the right-hand side of the inequality:[2sqrt{frac{s_1^2 + s_2^2}{10}} = 2sqrt{frac{0.036 + 0.04}{10}} = 2sqrt{frac{0.076}{10}} = 2sqrt{0.0076}]Now, let's calculate (sqrt{0.0076}). I know that (sqrt{0.0064} = 0.08) and (sqrt{0.0081} = 0.09), so (sqrt{0.0076}) should be between 0.08 and 0.09.Let me compute it more precisely. Let's see:(0.08^2 = 0.0064)(0.085^2 = 0.007225)(0.086^2 = 0.007396)(0.087^2 = 0.007569)(0.088^2 = 0.007744)So, 0.0076 is between (0.087^2) and (0.088^2). Let's approximate.The difference between 0.007569 and 0.0076 is 0.000031.The difference between 0.007744 and 0.007569 is 0.000175.So, 0.000031 / 0.000175 ≈ 0.177.So, approximately, (sqrt{0.0076} ≈ 0.087 + 0.177 times (0.088 - 0.087) ≈ 0.087 + 0.00177 ≈ 0.08877).So, roughly 0.0888.Therefore, (2 times 0.0888 ≈ 0.1776).So, the right-hand side is approximately 0.1776.Now, comparing (overline{y} - overline{x} = 0.3) to 0.1776.Since 0.3 > 0.1776, the inequality holds.Therefore, according to the given criterion, the mean of the indicator for the new device has significantly improved compared to the old device.Wait, but let me double-check my calculation for the square root. Maybe I can use a calculator method.Alternatively, I can compute (sqrt{0.0076}) as follows:Let me write 0.0076 as 76/10000.So, (sqrt{76/10000} = sqrt{76}/100).Now, (sqrt{76}) is between 8 and 9, since 8^2 = 64 and 9^2 = 81.Compute 8.7^2 = 75.698.7^2 = 75.698.71^2 = (8.7 + 0.01)^2 = 8.7^2 + 2*8.7*0.01 + 0.01^2 = 75.69 + 0.174 + 0.0001 = 75.86418.72^2 = 8.71^2 + 2*8.71*0.01 + 0.01^2 = 75.8641 + 0.1742 + 0.0001 = 76.0384So, (sqrt{76}) is between 8.71 and 8.72.Since 76 is 76.0000, and 8.71^2 = 75.8641, the difference is 76 - 75.8641 = 0.1359.Between 8.71 and 8.72, the difference in squares is 76.0384 - 75.8641 = 0.1743.So, 0.1359 / 0.1743 ≈ 0.78.Therefore, (sqrt{76} ≈ 8.71 + 0.78*(0.01) ≈ 8.71 + 0.0078 ≈ 8.7178).Therefore, (sqrt{76}/100 ≈ 8.7178/100 ≈ 0.087178).So, (sqrt{0.0076} ≈ 0.087178).Therefore, (2 * 0.087178 ≈ 0.174356).So, approximately 0.1744.Therefore, the right-hand side is approximately 0.1744.Comparing to (overline{y} - overline{x} = 0.3), which is greater than 0.1744.Therefore, the improvement is significant.Wait, but earlier I thought it was 0.1776, but with more precise calculation, it's approximately 0.1744. Either way, 0.3 is greater than both.So, yes, the difference is greater than the threshold, so the improvement is significant.Just to be thorough, let me compute (overline{y} - overline{x}) again:10.3 - 10.0 = 0.3.And the threshold:2 * sqrt((0.036 + 0.04)/10) = 2 * sqrt(0.076/10) = 2 * sqrt(0.0076) ≈ 2 * 0.087178 ≈ 0.174356.So, 0.3 > 0.174356, so yes, it's significant.Alternatively, another way to look at it is to compute the square of the difference and compare it to 4 * (s1^2 + s2^2)/10.So, (0.3)^2 = 0.09.4 * (0.036 + 0.04)/10 = 4 * 0.076 / 10 = 4 * 0.0076 = 0.0304.Since 0.09 > 0.0304, the difference is significant.Yes, that also confirms it.So, in conclusion, the mean of the indicator for the new device has significantly improved compared to the old device.**Final Answer**The mean of the indicator for the new device has significantly improved compared to the old device. (boxed{text{Significant Improvement}})

answer：Okay, so I have this problem here: Given that a is greater than b and both are positive, the quadratic function f(x) = ax² + 2x + b has exactly one zero. I need to find the minimum value of (a² + b²)/(a - b). The options are A:1, B:√2, C:2, D:2√2.Alright, let me start by understanding what it means for a quadratic function to have exactly one zero. I remember that for a quadratic equation ax² + bx + c = 0, the discriminant D is b² - 4ac. If D is zero, the quadratic has exactly one real root. So, in this case, the discriminant should be zero.Given f(x) = ax² + 2x + b, the coefficients are A = a, B = 2, and C = b. So, the discriminant D is B² - 4AC, which is 2² - 4*a*b = 4 - 4ab. Since there's exactly one zero, D must be zero. So, 4 - 4ab = 0. Let me solve this equation.4 - 4ab = 0 Subtract 4 from both sides: -4ab = -4 Divide both sides by -4: ab = 1.Okay, so ab equals 1. That's a key relationship between a and b. Since a > b > 0, both a and b are positive numbers, and a is larger than b.Now, the expression I need to minimize is (a² + b²)/(a - b). Let me write that down:E = (a² + b²)/(a - b)I need to find the minimum value of E given that ab = 1 and a > b > 0.Hmm, how can I approach this? Maybe I can express a in terms of b or vice versa since I know ab = 1. Let's try expressing a as 1/b because ab = 1 implies a = 1/b.So, substituting a = 1/b into E:E = ((1/b)² + b²)/(1/b - b)Let me compute the numerator and denominator separately.Numerator: (1/b² + b²) = (1 + b⁴)/b²Denominator: (1/b - b) = (1 - b²)/bSo, E becomes:E = [(1 + b⁴)/b²] / [(1 - b²)/b] = [(1 + b⁴)/b²] * [b/(1 - b²)] = (1 + b⁴)/(b(1 - b²))Hmm, that seems a bit complicated. Maybe I should try a different approach. Let's think about expressing a² + b² in terms of (a - b) and ab.I remember that a² + b² = (a - b)² + 2ab. Let me verify that:(a - b)² = a² - 2ab + b² So, (a - b)² + 2ab = a² - 2ab + b² + 2ab = a² + b². Yes, that works.So, a² + b² = (a - b)² + 2ab. Since ab = 1, this becomes:a² + b² = (a - b)² + 2*1 = (a - b)² + 2Therefore, E = (a² + b²)/(a - b) = [(a - b)² + 2]/(a - b)Let me split this fraction:E = (a - b)²/(a - b) + 2/(a - b) = (a - b) + 2/(a - b)So, E = (a - b) + 2/(a - b)Now, this looks like something I can apply the AM-GM inequality to. The AM-GM inequality states that for positive real numbers x and y, (x + y)/2 ≥ √(xy), with equality when x = y.Let me set x = (a - b) and y = 2/(a - b). Then, E = x + y.Applying AM-GM:(x + y)/2 ≥ √(xy) So, (x + y) ≥ 2√(xy)Substituting x and y:(a - b) + 2/(a - b) ≥ 2√[(a - b)*(2/(a - b))] Simplify the right side:√[(a - b)*(2/(a - b))] = √2So, (a - b) + 2/(a - b) ≥ 2√2Therefore, E ≥ 2√2So, the minimum value of E is 2√2, and this occurs when x = y, which means (a - b) = 2/(a - b). Let me solve for (a - b):(a - b) = 2/(a - b) Multiply both sides by (a - b):(a - b)² = 2 Take square roots:a - b = √2 (since a > b, a - b is positive)So, a - b = √2We also know that ab = 1.So, we have two equations:1. a - b = √2 2. ab = 1Let me solve for a and b.From equation 1: a = b + √2Substitute into equation 2:(b + √2)*b = 1 b² + √2 b - 1 = 0This is a quadratic equation in terms of b. Let me solve it using the quadratic formula:b = [-√2 ± √( (√2)² - 4*1*(-1) )]/(2*1) Simplify inside the square root:(√2)² = 2 So, √(2 + 4) = √6Thus,b = [-√2 ± √6]/2Since b > 0, we take the positive root:b = (-√2 + √6)/2Then, a = b + √2 = (-√2 + √6)/2 + √2 = (-√2 + √6 + 2√2)/2 = (√2 + √6)/2So, a = (√2 + √6)/2 and b = (-√2 + √6)/2Let me verify if ab = 1:a*b = [(√2 + √6)/2] * [(-√2 + √6)/2] Multiply the numerators:(√2 + √6)(-√2 + √6) = (√6)^2 - (√2)^2 = 6 - 2 = 4 So, a*b = 4/4 = 1. Correct.And a - b = √2, as required.So, the minimum value of E is indeed 2√2, achieved when a = (√2 + √6)/2 and b = (-√2 + √6)/2.Therefore, the answer is D: 2√2.**Final Answer**boxed{D}

answer：First, I need to find the principal amount (P) that, when invested at a simple interest rate of 9% per annum for 5 years, yields a total simple interest of 4016.25.I'll use the simple interest formula:[ text{Simple Interest (SI)} = frac{P times R times T}{100} ]Given:- SI = 4016.25- R = 9%- T = 5 yearsRearranging the formula to solve for P:[ P = frac{SI times 100}{R times T} ]Plugging in the values:[ P = frac{4016.25 times 100}{9 times 5} ][ P = frac{401625}{45} ][ P = 8925 ]So, the principal amount is 8925.

answer：Okay, so I need to find the average number of calls Jean answers per day. Hmm, let's see. She worked from Monday to Friday, right? That's five days in total. First, I should figure out how many calls she answered each day. On Monday, she answered 35 calls. That's straightforward. Then on Tuesday, she answered 46 calls. Okay, that's more than Monday. On Wednesday, she took 27 calls. That seems a bit less than the previous days. On Thursday, she answered 61 calls. Wow, that's a lot more! And finally, on Friday, she finished with 31 calls. So, to find the average, I think I need to add up all the calls she answered during the week and then divide that total by the number of days she worked. That makes sense because average is usually the total divided by the number of items or days in this case.Let me start by adding up the calls:Monday: 35 callsTuesday: 46 callsWednesday: 27 callsThursday: 61 callsFriday: 31 callsSo, adding them up: 35 + 46 + 27 + 61 + 31. Let me do this step by step to avoid mistakes.First, 35 plus 46. That should be 81.Then, 81 plus 27. That would be 108.Next, 108 plus 61. That adds up to 169.Finally, 169 plus 31. That gives me 200.Okay, so the total number of calls she answered during the week is 200.Now, since she worked five days, I need to divide the total calls by the number of days to find the average.So, 200 divided by 5. Let me calculate that.200 divided by 5 is 40.So, the average number of calls Jean answers per day is 40.Wait, let me double-check my addition to make sure I didn't make a mistake.35 + 46 is indeed 81.81 + 27 is 108.108 + 61 is 169.169 + 31 is 200.Yes, that seems correct.And dividing 200 by 5 definitely gives 40.So, I'm confident that the average number of calls Jean answers per day is 40.