course Mth 271
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17:53:56 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 need to find two positive numbers labeled x and y. I need the minimum sum of the first plus three times the second. Now I write my primary equation. S = 3x + y Now, I need a single independent variable. I know that the product of the two numbers is equal to 192. So I write the secondary equation xy = 192 and solve for y xy = 192 y = 192 / x I substitute this expression into the primary equation S = 3x + 192 / x Since both numbers are positive the desired domain will be 0 < x. Now I use calculus to find the minimum point. First I take the derivative. dS/dx = 3 - 192 / x^2 I set this equal to zero to find the critical point. 0 = 3 - 192 / x^2 192/x^2 = 3 3x^2 = 192 x^2 = 64 x = +-8 -8 does not work because it is not in the domain. I now find the second derivative and use the second derivative test to see if x = 8 is a minimum. dS/dx = 3 - 192/x^2 d^2S/dx = 384 / x^3 The second derivative is positive at 8 which means that the point is a minimum. confidence assessment: 3
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17:57:00 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 assessment: 3
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18:17:10 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 --> Problem #21 in my book begins with 16 trees, 80 apples per tree, and decreases 4 apples per tree so I used this information when I worked the problem. First, see that yield equals number of trees times apples per tree. I let Y equal yield, t equal trees and a equal apples per tree. Y = t*a Since for each additional tree beyond 16 the average yield decreases by 4, I wrote the equation Y = (16 + n)(80 - 4n) where n equals additional trees. I simplify Y = 1280 + 16n - 4n^2 Y = -n^2 + 4n + 320 The domain is 0
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18:17:18 How many trees should be planted and what will be the maximum yield?
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RESPONSE --> confidence assessment: 2
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18:19:54 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 --> I believe I worked the problem correctly for the information given in my book. I see that my domain was wrong, however, since I could have negative additional trees by subtracting from the existing 16. self critique assessment: 3
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18:21:13 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> I think it is amazing how I can take algebraic expressions and apply calculus to find the desured number of trees, dollars, etc.
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