Expert: Ahmed Salami - 4/25/2007

Question
1. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single strand electric fence. With 1600m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

2. Suppose that                                             is the cost of manufacturing items. Find a production level that will minimize the average cost of making  items.

3. There are 500 apple trees in an orchard. Each tree produces 8000 apples. For each additional tree planted in the orchard, the output per tree drops by 100 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees?  

Answer
Hi,
1)Considering that the rectangular fence would have only three sides made of wire, two of them would have to be identical. Let the length of each identical side be x and the remaining length be y, then the total length of the wire would be
L = 2x + y
The area of the fence would be
A = xy
We now have to find the maximum area possible with the constraint L = 2x + y
i.e 2x + y = 1600
Therefore,
y = 1600 - 2x
and
A = x(1600 - 2x)
 = 1600x - 2x^2
To determine the maximum area of A, we differentiate with respect to x and equate to zero. Hence,
dA/dx = 1600 - 4x
Equating to zero gives
1600 - 4x = 0
1600 = 4x
x = 400m
y = 1600 - 2(400)
 = 1600 - 800
 = 800m
The maximum area we can enclose is therefore
A = (400)(800)
 = 320000m^2

2)There's a part missing from this question but i'm going to assume its an equation of the form C = f(q) [cost as a function of quantity]
The average cost would be
AC = C/q
Average cost is minimised when
d(AC)/dq = 0
Good luck on that one!

3)The total output is the sum of the original output (500 x 8000) plus the output due to the x additional trees.
x additional trees, with each decreasing their output per tree by 100 apples, results in an average of 8000 - 100x apples per tree.
The total output is now (number of trees multiplied by average output)
Output = (500 x 8000) + x(8000 - 100x)
      = 4000000 + 8000x - 100x^2
The maximum output can be obtained  when we differentiate output with respect to x and equate to zero i.e
d(output)/dx = 8000 - 200x
equating to zero gives
8000 - 200x = 0
8000 = 200x
x = 40 trees
I hope i have understood your questions correctly and answered them accordingly.
You can always get back to me.
Regards.