a. How many trees should be planted per acre to maximize the total yield?

Question

1.   In a single growing season at the Smith Family Orchard, the average yield per apple tree is 150 apples when the number of trees per acre is 100. For each additional tree over 100, the average yield per tree decreases by 1.

a.   What would be the average yield per tree if the number of trees per acre was doubled? What would be the total yield in that case?

How many trees should be planted per acre to maximize the total yield?

Answer 1

Let y = yield per tree and x = # of trees

The y = 250-x

Total yield = xy = 250x - x²

^{Max yield occurs when derivative of total yield = 0, i.e. when 2x=250, i.e. x=125}