MinMax05 Yield of a field of crops.

An orange farmer knows from experience that in a certain field 60 orange trees will produce an average of 400 oranges per tree. For each additional tree planted the average yield per tree will drop by 4 oranges. What number of trees will produce maximum total yield?

The total yield for 60 trees with an average of 400 oranges per tree is:

Y|₆₀ = (60 trees) 400 oranges/tree = 24,000 oranges

For one more tree the yield is:

Y|₆₁ = (61 trees) 396 oranges/tree = 24,156 oranges

For another tree (total 62) the yield is:

Y|₆₂ = (62 trees) 392 oranges/tree = 24,304 oranges

Looking at these numbers, the general formula for total yield as the number of trees is increased is:

Y = (60 + x)(400 - 4x) where x is the number of trees in excess of 60.

Problem statements similar to this one can be confusing. You may have already figured that out! One way of getting a handle on the defining equation is to put in some numbers. In this case, writing the total yield for 60 trees producing an average of 400 oranges per tree and then increasing the number of trees by 1 and decreasing the yield per tree by 4, then repeating the process (increasing the number of trees to 62 and decreasing the yield per tree another 4 oranges) provides an education in how to write the general statement for the yield. The numbers also allow you to check the defining equation you have written.

Write the yield equation as Y = (60 + x)(400 - 4x) = 24,000 + 160x - 4x² .

The first derivative of Y is Y' = 160 - 8x and setting Y' = 0 , x = 20 .

The second derivative of Y is Y'' = -8 verifying that x = 20 is a maximum.

The total number of trees for maximum yield is 80 (20 more than the original 60).