Question

A farmer finds that if she plants 70 trees per acre, each tree will yield 75 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest?

Answer 1

Answer:

  44 3/8 trees per acre

Step-by-step explanation:

Let x represent the number of trees planted per acre. Let y represent the yield in bushels per tree. The farmer estimates the yield will be ...

  y = 75 -4(x -70)

  y = 355 -4x . . . . . . simplify

The yield per acre is this yield per tree multiplied by the number of trees per acre, x.

  yield per acre = x(355 -4x) = 4(x)(88.75 -x)

This describes a downward-opening parabola with zeros at x=0 and x=88.75. The vertex (maximum) is halfway between the zeros, at x = 44.375.

To maximize her harvest, the farmer should plant 44 3/8 trees per acre. If the model is correct, each tree would then produce 177.5 bushels of fruit, almost 2.4 times the current yield.