course MTH 271 12/15,1200 assignment #025025. `query 25
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20:17:44 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 What are the two desired numbers and how did you find them?
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RESPONSE --> xy=192 y=192/x x + 3y=S (Minimum) S= x + 3(192/x) =x + 576/x =x + 576x^-1 S'=1 + (-1)(576)x^-2 S'=1 - 576/x^2 (x^2 -576) / x^2 0=x^2 - 576 576=x^2 x=+-24 X=24 (-24 not an option as both numbers must be positive) y=192/x y=192/24 y=8 confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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20:17:55 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. **
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RESPONSE --> ------------------------------------------------ Self-critique Rating:ent: 3
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20:22:46 3.4.21 80 apple trees yield and average of 400 per tree; each additional tree decreases the yield by 32 apples per tree. Maximize yield.
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RESPONSE --> x=new trees S=80 + x Yield Per Tree=400 - 32x Total yield= x trees * apple yield/x trees (80 + x)(400-32x) 32000 - 2160x - 32x^2 Y=-32x^2 -2160x +32000 Y'=-64x -2160 0=-64x - 2160 2160=-64x -34=x S=80 + x S=80 +(-34) S=46 trees for maximum yield confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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20:24:06 How many trees should be planted and what will be the maximum yield?
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RESPONSE --> 46 trees should be planted confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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20:24:26 If we let x stand for the number of trees added to the 80 then the yield per tree is 400 - 32 x, and there would be 80 + x trees. The total yield would therefore be total yield = number of trees * yield per tree = (80 + x) * ( 400 - 32 x) = -32 x^2 + -2160 x + 32000, which is maximized when x = -34 approx.; this indicates -34 trees in addition to the 80, or 46 trees total. Another approach is to assume that only the additional trees experience the decrease. However it doesn't make sense for the yield decrease to apply only to the added trees and not to the original 400. If you're gonna crowd the orchard every tree should suffer. In any case, if we make the unrealistic assumption that the original 80 trees maintain their 400-apple-per-tree yield, and that the x additional trees each have a yield of 32 x below the 400, we have x added trees each producing 400 - 32 x apples, so we produce x (400 - 32 x) = 400 x - 32 x^2 additional apples. We therefore maximize the expression y = 400 x - 32 x^2. We obtain y ' = 400 - 64 x, which is 0 when -64 x + 400 = 0 or x = 6.25. Since y ' is positive for x < 6.25 and negative for x > 6.25 we see that 6.25 will be our maximizing value. We can't plant 6.25 trees, so the actual maximum must occur for either 6 or 7 trees. We easily see that the max occurs for 6 additional trees. So according to this interpretation we plant 86 trees. **
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RESPONSE --> ------------------------------------------------ Self-critique Rating:ent: 3
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20:24:59 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> It was surprising that the number of trees had to be reduced to maximize apple production. ------------------------------------------------ Self-critique Rating:ent: 3"