An almond grower finds that if she plants 29trees per acre each tree will yield

Question

An almond grower finds that if she plants 29trees per acre each tree will yield 82 bushels of almonds. She also estimated that for each additional tree that she plants per acre the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest and what is the maximum harvest? An almond grower finds that if she plants 29trees per acre each tree will yield 82 bushels of almonds. She also estimated that for each additional tree that she plants per acre the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest and what is the maximum harvest?

Explanation / Answer

he number of trees per acre will be 29 + x, where x is the number of additional trees planted.

The yield per tree is 82 - 2x.

The total harvest is:

(number of trees per acre)(yield per tree)

= (29 + x)(82 - 2x)
= 2378+24x-2x^2

To find the value of x which maximizes the yield, take the derivative of the yield function and set it equal to 0:

d/dx( 2378 + 24 x - 2x^2 ) = 0 ==> 24 - 4x = 0 ==> 4 x = 24 ==> x = 6

or

The maximum harvest occurs when x = -b/2a = - 24 /(2 * - 2) = 24 / 4 ==> 6

According to this, she should plant 29 + x = 29 + (6) = 35 trees per acre.
Number of trees should be ( 82 - 2(6) ) ==> 82 - 12 ==> 70

Harvest = 35 * 70 ==> 2450

The maximum harvest in this case is 2450 bushels per acre.

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