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.

 

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

 

 

025. `query 25

 

 

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:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

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?

 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

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: