#$&* course Mth 271 If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
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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): ------------------------------------------------ Self-critique Rating: ********************************************* 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: X + 80 = trees added and the yield would be 400-4x. Derive the equation for yield by foiling First Outet Inner Last (400 - 4 x)(80 + x) = = -4 x^2 + 80 x + 32000 Take the first derivative and get -8x + 80 and solve the equation for x and get x = 10 So you would have to plant an additional 10 trees giving the total of trees by 80 +x or 80 + 10 = 90. Yield would = 400-4X = 360 per tree 90 trees x 360 apples per tree = 32,400 apples. As your solution stated you cn plug back in to check solution &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ 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): ------------------------------------------------ Self-critique Rating: Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!