Find the number of trees per acre that will give the greatest yield per acre.

Question

An orchard has an average yield of 25 bushels per tree when there are at most 40 trees per acre. When there are more than 40 trees per acre, the average yield decreases by ½ bushel per tree for every tree over 40. Find the number of trees per acre that will give the greatest yield per acre.

Answer 1

x is number of trees per acre and y is yield in bushels per acre

For x<40 y=25x

For x≥40

y=(25-1/2(x-40))x

y=(25-1/2x+20)

y=-x²/2+45x

Maximum where derivative is zero or an end point

dy/dx=-x+45

0=-x+45

x=45

end points are x=40 or x=90

test these or do first or second derivative test

Test values

y=-(40)²/2+45(40)=1000

y=-(45)²/2+45(45)=1012.5

y=-(90)²/2+45(90)=0

Absolute max x=45

First derivative test:

test to the left and right of critical point to see if it is a max or min

less than 45 (but more than 40 because it must be in the domain)

x=43

dy/dx=-43+45=2 derivative is positive when x is less than 45

more than 45 (but less than 90 because it must be in the domain)

x=50

dy/dx=-50+45=-5 derivative is positive when x is more than 45

because the derivative is positive from 40 to 45 yield is increasing from x=40 to x=45

because the derivative is negative from 45 to 90 yield is increasing from x=45 to x=90

This means x=45 is a local max and since there are no other critical values then it is higher than the two endpoints

Second derivative test

d²y/dx²=-1

second derivative is negative so that means y is concave down so x=45 (the critical point above) is a local maximum. since there are no changes in concavity and only one critical value x=45 is the absolute max