Help Solving Calculus Problem

Question

An apple farm yields an average of 39 bushels of apples per tree when 21 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

Answer 1

This problem is a classic case of an Optimization. Optimization problems have an Objective function (The thing that is to be maximized or minimized) and certain constraints that limit the values of the Objective function. For this problem in particular, the Objective function would be the Yield per Acre. The constraint on the yield is the "crowding" effect from additional trees. Let's define the variables.

Y: Yield per Acre

B: Bushels per Tree

T: Trees per Acre

To calculate the Yield per acre, you would mutiply the Bushels per Tree by the Trees per Acre

Y=B*T

The bushels per tree is a function of the amount of trees per acre.

B=39 when T=21, B decreases by 1 for each increase in T. From this, we can create a point-slope form linear equation for B and T

B-B(21)=-1(T-21)

B-39=-T+21

B=-T+60

Frequently in Optimization problems, you can take a constraint and substitute it into the Objective function. This serves to reduce the number of independent variables in the Objective function

Y=(-T+60)*T

Y=-T^2+60T

Once at this stage with only 1 variable, you can take the derivative. This function is smooth, with a domain of T being between 0 and 60 inclusive (you can't produce a negative amount of apples). The relative maxima or minima for a smooth function will occur either at the edges of the domain or at some point within where the derivative is zero.

dY/dT=-2T+60

This gives the marginal yield as the number of trees planted changes. When this is zero, we reach a critical value.

0=-2T+60

2T=60

T=30

Thus, we can find the optimal operating conditions of planting 30 trees per acre. At this stage, we can also find the total yield.

Y=-T^2+60T

Y=-(30)^2+60(30)

Y=-900+1800

Y=900

We can produce a maximum of 900 apples per acre.

Answer 2

t represents the number of trees

Y = (39 - t)(21 + t)

Y = 819 + 18t - t²

Maximum occurs are -b/2a, or 9.

The highest yield is 30² or 900 bushels.