open qa 25

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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

What are the two desired numbers and how did you find them?

Your solution:

S = x + 3y

Xy = 192

S = x + 3 * 192/x = x + 576/x

x>0

S’ = 1 - 576/x^2

1 - 576/x^2 = 0

X^2 = 576

X = =- 24

x> 0

x = 24

y = 192/24 = 8

(24, 8)

Two intervals:

(0, 24) => decreasing

(24, infinity) => increasing

24 is the minimum.

confidence rating #$&*:

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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. **

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Self-critique (if necessary):

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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:

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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.

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The derivative is zero when -8 x + 80 = 0, so the solution x = 10 is the critical value.

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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..

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The yield per tree will be 400 - 10 * 4 = 360 so the total yield will be 90 trees * 360 apples / tree = 32 400 apples.

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If we evaluate the total-yield function -4 x^2 + 80 x + 32000 for x = 10, we get 32 400, verifying our solution.

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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.

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&#Good work. Let me know if you have questions. &#