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There are 50 apple trees in an orchard and each tree produces an average of 200 apples each year. For each additional tree planted within the orchard, the average number of Apple's produce drops by 5. What is the optimal number of trees to plant in the orchard?

Let x be the number of additional apple trees.

The number produced is then (200-5x)(50+x),

Multiplying this out gives the equation for the number of apples as

N(x) = 10,000 - 50x + 5x².

The derivative gives us N'(x) = -50 + 10x.

Setting that to 0 gives x = 5.

The variable x was for the number of trees, so that is the answer.

Calculus

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