#$&* course Mth 271 11/25 10p *********************************************
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Given Solution: `a First set up the primary equation S=x+3y (y being the 2nd number) and the secondary equation xy=192. So S = x + 3(192/x). We now maximize the function by finding critical points (points where the derivative is zero) and testing to see whether each gives a max, a min, or neither. S ' = 1 - 576 / x^2, which is zero when x = sqrt(576) = 24 (or -24, but the problem asks for positive numbers). For this value of x we get y = 192 / x = 192 / 24 = 8. So the numbers are x = 24 and y = 8. }Note that x = 24 does result in a min by the first derivative test, since S ' is negative for x < 24 and positive for x > 24. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I am having a hard time with these problems. I really couldn’t follow any of the problems I have attempted in the book. On this problem, it says above that x+3y gives a minimum, but x +3y turns out to be 24 +3(8) which is 48. The minimum that I tested and that matches your solution is 24. ------------------------------------------------ Self-critique Rating:1
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Given Solution: `a If we let x stand for the number of trees added to the 80 then the yield per tree is 400 - 4 x, and there would be 80 + x trees. The total yield is therefore total yield = yield per tree * number of trees = (400 - 4 x)(80 + x) = -4 x^2 + 80 x + 32000 The derivative of this function is - 8 x + 80, and the second derivative is -4. The derivative is zero when -8 x + 80 = 0, so the solution x = 10 is the critical value. The second derivative is negative, so a graph of the function is concave down, indicating that the critical value is a maximum. We conclude that the maximum yield is obtained by planting 20 additional trees, so that the total number of trees is 80 + 10 = 90.. The yield per tree will be 400 - 10 * 4 = 360 so the total yield will be 90 trees * 360 apples / tree = 32 400 apples. If we evaluate the total-yield function -4 x^2 + 80 x + 32000 for x = 10, we get 32 400, verifying our solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I had to look at the solution for this one, too, but I have several questions. It did not affect the outcome of the problem, but can you explain to me why the second derivative is -4? It seems that it should be -8. I agree that x= 10 is the critical value and the maximum, but I don’t understand why “ the maximum yield is obtained by planting 20 additional trees, so that the total number of trees is 80+10 = 90.” It seems like the maximum yield would be obtained by planting 10 trees. ------------------------------------------------ Self-critique Rating:1