Practice Exam 2, problem 10

The harvest per acre is the product of the number of trees per acre times the yield of each tree.

harvest= (number of trees per acre) ยท (yield of each tree)

Supppose we add x trees per acre. We have:

number of trees per acre= 10 +x
yield of each tree= 90 - 3x

The harvest is then given by

(10 + x) (90 - 3 x)

Expanding and simplifying:

h[x_] = 900 - 30 x + 90 x - 3 x^2

900 + 60 x - 3 x^2

To maximize this function we solve h'(x)=0

h '[x]

60 - 6 x == 0

The solution is x=10. This is a local maximum since
h''[x]=-6 is always negative. Since the graph of h[x] is an inverted
parabola, we know that this is indeed the absolute maximum of h[x]

Therefore, the farmer must plant 20 trees  per acre to maximize the harvest.

Converted by Mathematica  (March 12, 2003)