college precalculus question

Question

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?

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

What type of function did you use to model the scenario? Be specific.

Why did you choose that type of function? What characteristics of the model led you to make that choice?

Did you write a formula for the function? If not, do so now. Be sure to state what the independent and dependent variable represent in the model.

Answer 1

If I understand the problem, the Yield Y looks like this on a trending basis

Y = (100)(150)

= (100+1)(150-1)

= (100+2)(150-2) so the general trend will be if E is the number of extra trees

= (100+E)(150-E) = 15,000 + 50E - E²

Let's find the max yield, dY/dE = 50 - 2E =0 implies E = 25

This means that if you plant 25 extra trees, the Yield will be maximum at 15,625.

If you double the number of trees then, Y = (100+100)(150-100) = 10,000.

This should be enough information to answer all questions!