course Mth 271 ??????????§?}??assignment #025025. `query 25
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13:59:50 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 --> I started out by setting it up as xy=192 to solve for y then I divided through by x to set the function up as follows: y = 192/x then I set up the function for 3 times the second to get x + 3y When you fill in for y you get x + 3(192/x), then I multiplied through by x to get x^2 + 3(192) x^2 + 576 then I divided through by x^2 to get the minimum value as follows: ds/dx = 1-576/x^2=0 you can take the square root of eat side to get x^2 = 288
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14:02:09 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 --> I had the problem set up correctly, but I got 17 instead of 24 for x which in turn caused me to get the wrong answer for y. self critique assessment: 2
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14:07:46 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 --> To solve for how many trees will be needed I set the function up as follows: y-80 = 80-76/16-17 (x-16) y-80 = 4/-1 (x-16) y-80=4x + 64/-1 y-80 = -4x + 64 add 80 to each side you get y = -4x + 144 this gives you the function to solve for the maximum yield. Then you solve for x x(-4x + 144) -4x^2 + 144x -8x + 144 8x=144 divide through by 8 to get x = 18 which is the number of trees that need to be planted. Filling in x into the original function you get y=-4(18)+144 y=72 which is the maximum yield. confidence assessment: 2
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14:08:07 How many trees should be planted and what will be the maximum yield?
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RESPONSE --> 18 trees and the maximum yield is 72 confidence assessment: 2
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14:10:13 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 --> My thought process was a little different on this as you can see I got 18 trees to be planted and 72 as the max yield. I have copied your answer for further review. self critique assessment: 2
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14:11:13 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> I am going to go back and review the videos and the section in the book to reinterate what I have learned. These word problems can be tricky. self critique assessment: 2
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