Q:

# Two years ago your orange orchard contained 90 trees and the yield per tree was 80 bags of oranges. last year you removed 10 of the trees and noticed that the yield per tree increased to 85 bags. assuming that the yield per tree depends linearly on the number of trees in the orchard, what should you do this year to maximize your total yield?

Accepted Solution

A:
Answer:69 trees must be there this year to get the maximum yield.Step-by-step explanation:Let the number of trees = x.For 90 trees, the total yield is 80 bags.That is, the yield per tree = $$\frac{80}{90}=0.88$$For 80 trees, the total yield is 85 bags.That is, the yield per tree = $$\frac{85}{80}=1.06$$So, For every decrease in 10 trees, the yield per trees increases by 1.06-0.88 = 0.18.Thus, yield per tree, $$Y=1.06-\frac{x-80}{10}\times 0.18$$i.e. Yield per tree = $$Y=2.5-0.018x$$Then, the total yield, T = Number of trees(x) × Yield per tree(Y)i.e. Total yield, T = $$x(2.5-0.018x)$$i.e. Total yield, T = $$2.5x-0.018x^2$$Now, to maximize the total yield, we will differentiate T with respect to x and equate to 0.We get, $$T'=0$$ implies $$2.5-0.036x=0$$i.e. $$2.5=0.036x$$i.e. x= 69Thus, 69 trees must be there this year to get the maximum yield.