#$&* course Mth 271 Question: `q Problem 1 b 7th edition 3.4.6 find two positive numbers such that the product is 192 and a sum of the first plus three times the second is a minimum
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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): ********************************************* Question: `q Problem 5 80 apple trees in a certain field will yield an average of 400 per tree; each additional tree decreases the yield by 4 apples per tree. How many trees should be planted to maximize the yield? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ------------------------------------------------ Self-critique Rating:
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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 didn’t come up with this one on my own. It all makes perfect sense when I read the solution, but it’s hard to put it all together. These types of problems are still a challenge. ------------------------------------------------ Self-critique Rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!