calculus word problem

Question

An apple orchard has 2000 trees per hectare and the average yield is 300 apples per tree. For each additional tree planted per hectare the average yield per tree is reduced by about 0.1 apple. How many trees per hectare will give the largest crop of apples?

Answer 1

Here is a sketch of the solution. Please leave a comment if you have any questions.

Let x be the number of trees in excess of 2,000 trees per hectare. Let T be the total number of trees per hectare and let Y be the total yield per hectare. If you think about what the problem says, you should come to the conclusion that

T = 2000 + x,

Y = (300 - 0.1 x) T = (300 - 0.1 x)(2000 + x).

Now differentiate Y with respect to x and find the value of x for which Y'(x) = 0. The answer is then 2000 + x.

Note that you should verify that Y''(x) < 0 to ensure that Y(x) is a maximum.