answer：Okay, so I have this problem about water flow in an urban system modeled by a quadratic equation. The equation given is f(x) = -2x² + 8x + 5, where x is the time in hours since the system started. There are two parts to the problem: first, finding the time when the water flow is maximum and the corresponding flow rate, and second, determining the interval when the flow rate stays above 9 cubic meters per hour.Starting with the first part: finding the maximum flow rate. Since the equation is quadratic, I remember that the graph of a quadratic function is a parabola. The coefficient of x² is -2, which is negative, so the parabola opens downward. That means the vertex of the parabola is the highest point, which corresponds to the maximum flow rate.To find the vertex of a quadratic function in standard form, f(x) = ax² + bx + c, the x-coordinate of the vertex is given by -b/(2a). In this equation, a is -2 and b is 8. So plugging those into the formula: x = -8/(2*(-2)) = -8/(-4) = 2. So the time when the maximum flow occurs is at x = 2 hours.Now, to find the maximum flow rate, I need to plug x = 2 back into the original equation. Let's compute that:f(2) = -2*(2)² + 8*(2) + 5= -2*4 + 16 + 5= -8 + 16 + 5= (16 - 8) + 5= 8 + 5= 13So the maximum flow rate is 13 cubic meters per hour at 2 hours after the system started.Moving on to the second part: finding the interval when the flow rate is above 9 cubic meters per hour. This means I need to solve the inequality f(x) > 9.So, starting with the equation:-2x² + 8x + 5 > 9Subtract 9 from both sides to set it to zero:-2x² + 8x + 5 - 9 > 0-2x² + 8x - 4 > 0Let me simplify this inequality. First, I can factor out a common factor of -2 to make it easier:-2(x² - 4x + 2) > 0But dealing with the negative coefficient might complicate things, so maybe I'll multiply both sides by -1 to make the coefficient positive. However, I remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. So:-2x² + 8x - 4 > 0Multiply both sides by -1:2x² - 8x + 4 < 0Now, let's see if I can factor this quadratic or if I need to use the quadratic formula. The quadratic is 2x² - 8x + 4. Let me try factoring:First, factor out a 2: 2(x² - 4x + 2). Hmm, x² - 4x + 2 doesn't factor nicely because the discriminant is (-4)² - 4*1*2 = 16 - 8 = 8, which is not a perfect square. So, I need to use the quadratic formula to find the roots.The quadratic formula is x = [-b ± sqrt(b² - 4ac)]/(2a). For the equation x² - 4x + 2 = 0, a = 1, b = -4, c = 2.So,x = [4 ± sqrt(16 - 8)]/2= [4 ± sqrt(8)]/2= [4 ± 2*sqrt(2)]/2= 2 ± sqrt(2)So the roots are x = 2 + sqrt(2) and x = 2 - sqrt(2). Since sqrt(2) is approximately 1.414, these roots are approximately 2 + 1.414 = 3.414 and 2 - 1.414 = 0.586.So, the quadratic 2x² - 8x + 4 factors as 2(x - (2 + sqrt(2)))(x - (2 - sqrt(2))). Since the coefficient of x² is positive, the parabola opens upward. Therefore, the quadratic is below zero (negative) between its two roots.So, 2x² - 8x + 4 < 0 when x is between 2 - sqrt(2) and 2 + sqrt(2). Therefore, the original inequality -2x² + 8x - 4 > 0 is satisfied for x between 2 - sqrt(2) and 2 + sqrt(2).But wait, let me double-check. The original inequality after moving 9 to the left was -2x² + 8x - 4 > 0. Then, multiplying by -1 gave 2x² - 8x + 4 < 0, which is true between the roots. So yes, the solution is x between 2 - sqrt(2) and 2 + sqrt(2).So, the interval during which the flow rate is above 9 cubic meters per hour is from approximately 0.586 hours to approximately 3.414 hours.But since the question asks for the interval, I should present it in exact terms rather than decimal approximations. So, the interval is (2 - sqrt(2), 2 + sqrt(2)).To recap:1. The maximum flow occurs at x = 2 hours, with a flow rate of 13 cubic meters per hour.2. The flow rate remains above 9 cubic meters per hour from 2 - sqrt(2) hours to 2 + sqrt(2) hours after the system starts.I think that's all. Let me just make sure I didn't make any calculation errors.For the vertex: x = -b/(2a) = -8/(2*(-2)) = 2. Plugging back in: -2*(4) + 16 + 5 = -8 + 16 + 5 = 13. Correct.For the inequality: starting from f(x) > 9, leading to -2x² +8x -4 >0, then multiplying by -1 to get 2x² -8x +4 <0. Factoring gives roots at 2 ± sqrt(2). Since the quadratic opens upwards, the inequality is satisfied between the roots. So interval is (2 - sqrt(2), 2 + sqrt(2)). Correct.Yes, that seems right.**Final Answer**1. The maximum flow occurs at boxed{2} hours with a flow rate of boxed{13} cubic meters per hour.2. The flow rate remains above 9 cubic meters per hour during the interval boxed{(2 - sqrt{2}, 2 + sqrt{2})} hours.

answer：Alright, so I have this problem here about a semicircular region where the perimeter is 20. They mention that the perimeter includes both the semicircular arc and the diameter. I need to find the area of this region, and the options are given as (A) 36.6, (B) 23.8, (C) 49.3, (D) 51.6, and (E) 26.7. The question is asking which one it's closest to.Hmm, okay. Let me think about this step by step. First, I should recall the formulas related to a semicircle. I remember that the perimeter (or circumference) of a semicircle isn't just half the circumference of a full circle; it also includes the diameter. So, the formula for the perimeter (P) of a semicircle would be the length of the curved part plus the diameter.Let me write that down. The circumference of a full circle is 2πr, so half of that would be πr. Then, adding the diameter, which is 2r, gives the total perimeter. So, P = πr + 2r.In this problem, the perimeter is given as 20. So, I can set up the equation:πr + 2r = 20.I need to solve for r, the radius. Let me factor out the r:r(π + 2) = 20.So, r = 20 / (π + 2).Okay, now I can compute the value of r. Let me approximate π as 3.1416 for better accuracy.So, π + 2 is approximately 3.1416 + 2 = 5.1416.Therefore, r = 20 / 5.1416 ≈ 20 / 5.1416.Let me calculate that. 20 divided by 5.1416. Hmm, 5.1416 times 3 is about 15.4248, and 5.1416 times 3.8 is roughly 5.1416*3=15.4248 plus 5.1416*0.8=4.1133, so total is approximately 19.5381. That's close to 20, so maybe 3.8 is a good approximation.Wait, let me do it more accurately. 5.1416 * 3.8 = ?Well, 5 * 3.8 = 19, and 0.1416 * 3.8 ≈ 0.538. So total is approximately 19.538, as I thought earlier. So 5.1416 * 3.8 ≈ 19.538, which is just a bit less than 20. The difference is 20 - 19.538 = 0.462.So, to get the exact value, how much more than 3.8 is needed? Let me see, 0.462 / 5.1416 ≈ 0.09. So, 3.8 + 0.09 ≈ 3.89. So, r ≈ 3.89.Wait, let me check that. 5.1416 * 3.89: 5 * 3.89 = 19.45, and 0.1416 * 3.89 ≈ 0.550. So, total is approximately 19.45 + 0.55 = 20.00. Perfect, so r ≈ 3.89.So, the radius is approximately 3.89 units.Now, moving on to the area. The area of a full circle is πr², so the area of a semicircle would be half of that, which is (1/2)πr².So, plugging in r ≈ 3.89, let's compute the area.First, compute r squared: 3.89². Let me calculate that. 3.89 * 3.89.3 * 3 = 9, 3 * 0.89 = 2.67, 0.89 * 3 = 2.67, and 0.89 * 0.89 ≈ 0.7921.Wait, maybe a better way is to compute 3.89 * 3.89:First, 3 * 3 = 9.3 * 0.89 = 2.67.0.89 * 3 = 2.67.0.89 * 0.89 ≈ 0.7921.Wait, no, that's not the right way. Let me do it properly.3.89 * 3.89:Multiply 3.89 by 3.89.Let me write it as:3.89x3.89--------Multiply 3.89 by 9: 35.01Multiply 3.89 by 80 (shift one position): 311.2Multiply 3.89 by 300 (shift two positions): 1167Now, add them up:35.01311.21167--------Total: 35.01 + 311.2 = 346.21; 346.21 + 1167 = 1513.21Wait, that can't be right because 3.89 squared is about 15.1321, not 1513.21. I must have messed up the decimal places.Wait, no, actually, 3.89 is approximately 4, so 4 squared is 16, so 3.89 squared should be a little less than 16, say around 15.13.Wait, let me do it more carefully.3.89 * 3.89:First, 3 * 3 = 9.3 * 0.89 = 2.67.0.89 * 3 = 2.67.0.89 * 0.89 ≈ 0.7921.Wait, perhaps I should use the formula (a + b)² = a² + 2ab + b², where a = 3 and b = 0.89.So, (3 + 0.89)² = 3² + 2*3*0.89 + 0.89² = 9 + 5.34 + 0.7921 ≈ 9 + 5.34 = 14.34 + 0.7921 ≈ 15.1321.Yes, that's correct. So, 3.89 squared is approximately 15.1321.So, r² ≈ 15.1321.Now, the area of the semicircle is (1/2)πr² ≈ 0.5 * π * 15.1321.Compute that: 0.5 * π ≈ 1.5708.So, 1.5708 * 15.1321 ≈ ?Let me compute 1.5708 * 15 = 23.562, and 1.5708 * 0.1321 ≈ 0.207.So, total area ≈ 23.562 + 0.207 ≈ 23.769.So, approximately 23.77.Looking at the options, (B) is 23.8, which is very close to 23.77. So, the area is closest to 23.8.Wait, but let me double-check my calculations because sometimes approximations can lead to errors.First, let me recalculate r:Perimeter P = πr + 2r = 20.So, r = 20 / (π + 2) ≈ 20 / (3.1416 + 2) ≈ 20 / 5.1416 ≈ 3.89.Yes, that's correct.Then, r² ≈ 3.89² ≈ 15.1321.Area = (1/2)πr² ≈ 0.5 * 3.1416 * 15.1321.Compute 0.5 * 3.1416 = 1.5708.1.5708 * 15.1321.Let me compute 1.5708 * 15 = 23.562.1.5708 * 0.1321: Let's compute 1.5708 * 0.1 = 0.15708, 1.5708 * 0.03 = 0.047124, 1.5708 * 0.0021 ≈ 0.0033.So, adding those: 0.15708 + 0.047124 ≈ 0.2042 + 0.0033 ≈ 0.2075.So, total area ≈ 23.562 + 0.2075 ≈ 23.7695, which is approximately 23.77.So, 23.77 is very close to 23.8, which is option (B). So, that should be the answer.But just to make sure, let me compute the area using a more precise value of r.Alternatively, maybe I can compute it without approximating r so early.Let me try that.Given that r = 20 / (π + 2).So, area = (1/2)πr² = (1/2)π*(20 / (π + 2))².Let me compute that expression step by step.First, compute (20 / (π + 2))².Which is 400 / (π + 2)².So, area = (1/2)π * (400 / (π + 2)²) = (200π) / (π + 2)².Now, let's compute this value numerically.First, compute π + 2 ≈ 3.1416 + 2 = 5.1416.So, (π + 2)² ≈ (5.1416)² ≈ 26.433.Compute 200π ≈ 200 * 3.1416 ≈ 628.32.So, area ≈ 628.32 / 26.433 ≈ ?Compute 628.32 divided by 26.433.Let me see: 26.433 * 23 = 26.433 * 20 = 528.66, plus 26.433 *3 = 79.299, total ≈ 528.66 + 79.299 ≈ 607.959.26.433 * 24 = 607.959 + 26.433 ≈ 634.392.But 628.32 is between 26.433*23 and 26.433*24.Compute 628.32 - 607.959 ≈ 20.361.So, 20.361 / 26.433 ≈ 0.77.So, total area ≈ 23 + 0.77 ≈ 23.77.So, same result as before, approximately 23.77, which is 23.8 when rounded to one decimal place.Therefore, the area is closest to 23.8, which is option (B).Wait, but let me check if I made any mistake in the calculations.Wait, when I computed (π + 2)², I approximated it as 26.433. Let me compute it more accurately.(π + 2)² = π² + 4π + 4.Compute π² ≈ 9.8696.4π ≈ 12.5664.So, total is 9.8696 + 12.5664 + 4 ≈ 26.436.Yes, so that's correct.200π ≈ 628.3185.So, 628.3185 / 26.436 ≈ ?Let me compute 26.436 * 23 = 26.436*20=528.72, 26.436*3=79.308, total 528.72 +79.308=608.028.Subtract from 628.3185: 628.3185 - 608.028 =20.2905.Now, 20.2905 /26.436 ≈ 0.767.So, total area ≈23 +0.767≈23.767, which is approximately 23.77, same as before.So, 23.77 is very close to 23.8, so option (B) is correct.But just to be thorough, let me check if perhaps I made a mistake in the initial formula.Wait, the perimeter of a semicircle is πr + 2r, right? Because it's half the circumference plus the diameter.Yes, that's correct.So, if the perimeter is 20, then πr + 2r =20, so r=20/(π+2).So, that's correct.Then, area is (1/2)πr², which is correct.So, I think my calculations are correct.Wait, but let me just compute 20/(π+2) more accurately.π ≈3.1415926536.So, π +2≈5.1415926536.20 divided by 5.1415926536.Let me compute that.5.1415926536 *3=15.4247779608.Subtract from 20: 20 -15.4247779608≈4.5752220392.Now, 5.1415926536 *0.8=4.1132741229.Subtract from 4.5752220392: 4.5752220392 -4.1132741229≈0.4619479163.Now, 5.1415926536 *0.09≈0.4627433388.So, 0.4619479163 is slightly less than 0.4627433388.So, 0.09 - (0.4627433388 -0.4619479163)/5.1415926536.Wait, maybe it's easier to compute 0.4619479163 /5.1415926536≈0.09 approximately.So, total r≈3 +0.8 +0.09≈3.89, as before.So, r≈3.89.So, that's correct.Therefore, the area is approximately 23.77, which is closest to 23.8, which is option (B).Wait, but let me just check the options again: (A) 36.6, (B) 23.8, (C)49.3, (D)51.6, (E)26.7.So, 23.77 is closest to 23.8, so (B). That seems correct.Wait, but just to make sure, maybe I can compute the area using a calculator-like approach.Alternatively, perhaps I can compute it using fractions.Wait, but maybe I can compute it more accurately.Wait, let me compute r as 20/(π+2). Let me compute π as 3.1415926536.So, π+2=5.1415926536.20 divided by 5.1415926536 is approximately 3.8915.So, r≈3.8915.Then, r squared is (3.8915)^2.Compute 3.8915*3.8915.Let me compute 3*3=9, 3*0.8915=2.6745, 0.8915*3=2.6745, and 0.8915*0.8915≈0.7948.Wait, but that's not the right way to compute it. Let me compute it properly.Compute 3.8915 * 3.8915:First, compute 3 * 3.8915 =11.6745.Then, 0.8 *3.8915=3.1132.0.09 *3.8915≈0.3502.0.0015*3.8915≈0.0058.Wait, no, that's not the right way. Alternatively, perhaps I should compute 3.8915 *3.8915 as:(3 + 0.8915)^2 = 3^2 + 2*3*0.8915 + (0.8915)^2.So, 9 + 5.349 + 0.7948≈9 +5.349=14.349 +0.7948≈15.1438.So, r squared≈15.1438.Then, area is (1/2)*π*r²≈0.5*3.1415926536*15.1438.Compute 0.5*3.1415926536≈1.5707963268.Multiply by 15.1438:1.5707963268 *15≈23.561944902.1.5707963268 *0.1438≈0.2258.So, total area≈23.561944902 +0.2258≈23.7877.So, approximately 23.7877, which is about 23.79, which is still closest to 23.8.Therefore, the answer is (B) 23.8.I think that's solid. I don't see any mistakes in my calculations.**Final Answer**boxed{B}

answer：Okay, so I have this problem about a front-end designer trying to optimize a web application by balancing two security algorithms, A and B. The problem has two parts, both involving quadratic equations. Let me try to figure them out step by step.Starting with part 1: Algorithm A's efficiency is modeled by the quadratic equation ( f(x) = 2x^2 - 8x + 5 ). I need to find the number of hours ( x ) the designer should spend to achieve the minimum execution time for Algorithm A.Hmm, quadratic equations. I remember that the graph of a quadratic equation is a parabola. Since the coefficient of ( x^2 ) is positive (which is 2 in this case), the parabola opens upwards. That means the vertex of the parabola is the minimum point. So, the minimum execution time occurs at the vertex of this quadratic function.To find the vertex of a quadratic function in standard form ( ax^2 + bx + c ), the x-coordinate is given by ( x = -frac{b}{2a} ). Let me apply that here.In this equation, ( a = 2 ) and ( b = -8 ). Plugging into the formula:( x = -frac{-8}{2 times 2} = frac{8}{4} = 2 ).So, the minimum execution time occurs at ( x = 2 ) hours. That seems straightforward. Let me just double-check my calculations. Yes, ( -8 ) divided by ( 4 ) is indeed ( 2 ). So, the designer should spend 2 hours optimizing Algorithm A for the minimum execution time.Moving on to part 2: Algorithm B's execution time is given by the inequality ( 3x^2 - 12x + 10 < 0 ). The designer needs to find the range of hours ( x ) that satisfy this condition to ensure the system remains secure.Alright, so I need to solve the inequality ( 3x^2 - 12x + 10 < 0 ). Since this is a quadratic inequality, I should first find the roots of the corresponding quadratic equation ( 3x^2 - 12x + 10 = 0 ). The sign of the quadratic expression will depend on the intervals determined by these roots.Let me try to solve the quadratic equation. The quadratic formula is ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ). Here, ( a = 3 ), ( b = -12 ), and ( c = 10 ).Calculating the discriminant first: ( b^2 - 4ac = (-12)^2 - 4 times 3 times 10 = 144 - 120 = 24 ).So, the discriminant is 24, which is positive, meaning there are two real roots. Let's compute them:( x = frac{-(-12) pm sqrt{24}}{2 times 3} = frac{12 pm sqrt{24}}{6} ).Simplify ( sqrt{24} ). Since ( 24 = 4 times 6 ), ( sqrt{24} = 2sqrt{6} ). So,( x = frac{12 pm 2sqrt{6}}{6} ).We can simplify this further by dividing numerator and denominator by 2:( x = frac{6 pm sqrt{6}}{3} ).Which can be written as:( x = 2 pm frac{sqrt{6}}{3} ).So, the two roots are ( x = 2 + frac{sqrt{6}}{3} ) and ( x = 2 - frac{sqrt{6}}{3} ).Let me approximate ( sqrt{6} ) to get a sense of the numerical values. ( sqrt{6} ) is approximately 2.449. So,( 2 + frac{2.449}{3} approx 2 + 0.816 = 2.816 )and( 2 - frac{2.449}{3} approx 2 - 0.816 = 1.184 ).So, the roots are approximately 1.184 and 2.816.Since the quadratic coefficient ( a = 3 ) is positive, the parabola opens upwards. That means the quadratic expression ( 3x^2 - 12x + 10 ) is below zero (negative) between its two roots. Therefore, the solution to the inequality ( 3x^2 - 12x + 10 < 0 ) is the interval between the smaller root and the larger root.So, the range of ( x ) that satisfies the inequality is ( 2 - frac{sqrt{6}}{3} < x < 2 + frac{sqrt{6}}{3} ).Let me write that in exact terms:( frac{6 - sqrt{6}}{3} < x < frac{6 + sqrt{6}}{3} ).Alternatively, simplifying:( 2 - frac{sqrt{6}}{3} < x < 2 + frac{sqrt{6}}{3} ).Just to make sure I didn't make a mistake, let me test a value within the interval and outside to see if the inequality holds.Let's pick ( x = 2 ), which is between 1.184 and 2.816.Plugging into ( 3x^2 - 12x + 10 ):( 3(4) - 12(2) + 10 = 12 - 24 + 10 = -2 ), which is less than 0. So, that works.Now, let's pick ( x = 0 ), which is less than 1.184.( 3(0) - 12(0) + 10 = 10 ), which is greater than 0. So, the inequality doesn't hold here, which is correct.Similarly, pick ( x = 3 ), which is greater than 2.816.( 3(9) - 12(3) + 10 = 27 - 36 + 10 = 1 ), which is greater than 0. So, the inequality doesn't hold here either, which is correct.Therefore, my solution seems correct.So, summarizing:1. For Algorithm A, the minimum execution time occurs at ( x = 2 ) hours.2. For Algorithm B, the execution time is below the threshold when ( x ) is between ( 2 - frac{sqrt{6}}{3} ) and ( 2 + frac{sqrt{6}}{3} ) hours.I think that's all. Let me just write the final answers clearly.**Final Answer**1. The designer should spend boxed{2} hours to achieve the minimum execution time for Algorithm A.2. The range of hours ( x ) that satisfy the condition for Algorithm B is boxed{left(2 - dfrac{sqrt{6}}{3}, 2 + dfrac{sqrt{6}}{3}right)}.

answer：Okay, so I have two problems here that I need to solve. Let me take them one at a time.Starting with the first problem: A graphic design student is working on a zine with a series of poems. Each page has a unique layout combining text and graphics. The student wants the ratio of pages with more graphics to pages with more text to be 3:2. The total number of pages is x, and there are 20 pages with more text. I need to set up and solve a linear equation to find the total number of pages in the zine.Alright, let's break this down. The ratio of graphics-heavy pages to text-heavy pages is 3:2. That means for every 3 pages with more graphics, there are 2 pages with more text. So, if I let the number of graphics-heavy pages be 3k and text-heavy pages be 2k, where k is some constant, then the total number of pages x would be 3k + 2k = 5k.But wait, the problem says there are 20 pages with more text. So, 2k = 20. That should help me find k. Let me write that down:2k = 20To solve for k, I can divide both sides by 2:k = 20 / 2k = 10So, k is 10. That means the number of graphics-heavy pages is 3k = 3*10 = 30 pages. Therefore, the total number of pages x is 30 + 20 = 50 pages.Wait, let me make sure I didn't skip any steps. The ratio is 3:2, so graphics:text = 3:2. If text pages are 20, then 2 parts correspond to 20 pages. So each part is 10 pages. Therefore, graphics pages are 3 parts, which is 30. Total pages 30 + 20 = 50. Yeah, that seems right.So, the equation I set up was based on the ratio. Let me represent this as a linear equation. Let me denote the number of graphics-heavy pages as G and text-heavy pages as T. Given that G/T = 3/2, and T = 20. So, G = (3/2)*T. Plugging in T = 20, G = (3/2)*20 = 30. Therefore, total pages x = G + T = 30 + 20 = 50.Alternatively, if I set up the equation in terms of x, since the ratio is 3:2, the total parts are 5. So, each part is x/5. Therefore, text-heavy pages are 2*(x/5) = 20. So, 2x/5 = 20. Solving for x:2x/5 = 20Multiply both sides by 5: 2x = 100Divide both sides by 2: x = 50Yes, that also gives x = 50. So, either way, the total number of pages is 50.Alright, moving on to the second problem. The student wants to create a special fold-out page that displays a poem surrounded by a spiral graphic. The area of the fold-out page is modeled by the quadratic expression A = x² - 4x - 5 square inches, where x is the length of one side of the square fold-out page. I need to find the dimensions of the fold-out page by solving the quadratic equation for when the area is zero and interpret the solution.Hmm, okay. So, the area A is given by A = x² - 4x - 5. They want to find the dimensions when the area is zero. So, set A = 0 and solve for x.So, 0 = x² - 4x - 5.I need to solve this quadratic equation. Let me write it as:x² - 4x - 5 = 0I can try factoring this. Looking for two numbers that multiply to -5 and add up to -4. Let's see, factors of -5 are 1 and -5 or -1 and 5.1 and -5: 1 + (-5) = -4. Perfect.So, the equation factors as:(x + 1)(x - 5) = 0Setting each factor equal to zero:x + 1 = 0 => x = -1x - 5 = 0 => x = 5So, the solutions are x = -1 and x = 5.But x represents the length of one side of the square fold-out page. Since length can't be negative, x = -1 doesn't make sense in this context. Therefore, the only valid solution is x = 5 inches.So, the dimensions of the fold-out page are 5 inches by 5 inches, since it's a square.Wait, let me make sure I didn't make a mistake in factoring. Let me expand (x + 1)(x - 5):x*x = x²x*(-5) = -5x1*x = x1*(-5) = -5So, combining like terms: x² -5x + x -5 = x² -4x -5. Yes, that matches the original equation. So, factoring was correct.Alternatively, I could have used the quadratic formula. Let me try that to double-check.Quadratic formula: x = [4 ± sqrt( (-4)^2 - 4*1*(-5) )]/(2*1)Calculating discriminant: 16 - (-20) = 16 + 20 = 36So, sqrt(36) = 6Therefore, x = [4 ±6]/2So, x = (4 + 6)/2 = 10/2 = 5x = (4 -6)/2 = (-2)/2 = -1Same solutions. So, x = 5 or x = -1. Again, x = 5 is the valid solution.Interpreting the solution: When the area A is zero, the length of the side is 5 inches. But wait, area being zero would imply that the page doesn't exist physically, right? Because area zero would mean it's not there. So, maybe the equation models the area as a function of x, and when x is 5, the area is zero, which might indicate a specific point where the spiral graphic starts or ends? Hmm, not entirely sure about the interpretation, but mathematically, x = 5 is the only feasible solution.Alternatively, maybe the quadratic models the area as a function, and when x is 5, the area is zero, which could be a boundary condition or something. But since x is the length of the side, and it's a square, the only meaningful solution is 5 inches.So, summarizing:1. The total number of pages in the zine is 50.2. The fold-out page has dimensions 5 inches by 5 inches.**Final Answer**1. The total number of pages in the zine is boxed{50}.2. The dimensions of the fold-out page are boxed{5} inches by boxed{5} inches.

answer：First, I need to establish the relationship between donations and ticket sales. Donations should be at least 25% of the total ticket sales. If each ticket is priced at ( t ) dollars and ( n ) tickets are sold, the total ticket sales amount to ( t times n ). Therefore, donations ( d ) should satisfy the inequality ( d geq 0.25 times t times n ).Next, the total funds raised, which is the sum of total ticket sales and donations, needs to be at least 5000. This gives the inequality ( t times n + d geq 5000 ).Given that ( t = 30 ) dollars and donations are exactly 25% of the total ticket sales, I can substitute ( d = 0.25 times 30 times n ) into the total funds inequality. This simplifies to ( 30n + 7.5n geq 5000 ), which further simplifies to ( 37.5n geq 5000 ). Solving for ( n ) gives ( n geq frac{5000}{37.5} ), resulting in ( n geq 133.overline{3} ). Since the number of tickets must be a whole number, the minimum number of tickets that need to be sold is 134.

answer：First, I need to solve the quadratic equation ( n^2 - 10n + 21 = 0 ) to determine the number of servers running Windows Server 2003 that need replacement. I'll use the quadratic formula for this.Next, after finding the number of outdated servers, I'll subtract that from the total number of servers (24) to find out how many servers will remain after the replacement. Then, I'll use the information that each new Linux server can handle twice the workload compared to the previous system and that the combined workload of the remaining servers is 192 units to determine the workload each new Linux server can handle.