Question

A farmer finds that if she plants 85 trees per acre, each tree will yield 80 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:

52 Trees

Step-by-step explanation:

If she plants 85 trees per acre, each tree will yield 80 bushels of fruit.

Yield = 85 X 80

Let x be the additional tree planted per acre.

For each additional tree planted per acre, the yield of each tree will decrease by 4 bushels.

Therefore, the yield:

$F(x)=(85+x)(80-4x)\\F(x)=6800-260x-4x^2$

Next, we maximize F(x) by taking its derivatives and solving for its critical point.

$F'(x)=-260-8x\\ Setting F'(x)=0, we have: \\-260-8x=0\\8x=-260\\ Divide both sides by 8 \\x=-32.5$

$F''(x)=-8, therefore we have a maximum$

It is apparent that she currently has too many trees per acre. To get a maximum harvest, she needs to reduce the number of trees by approximately 32.5 trees.

Therefore, number of trees per acre she should plant:

85+(-32.5)

=85-32.5

=52.5 Trees

Since the number of trees can only be an integer and it cannot be greater than 52.5, the number of trees to be planted per acre to maximize her harvest is 52.