MHB BAM's question at Yahoo Answers regarding maximizing the yield of an orchard

[MarkFL]

Here is the question:

Help with calculus homework?


Hello! I am having much difficulty this with problem. I just don't get it. Help would be appreciated. Thanks very much! 5 points for correct answer ! :)

An orchard contains 61 peach trees with each tree yielding an average of 56 peaches. For each 4 additional trees planted, the average yield per tree decreases by 9 peaches. How many trees should be planted to maximize the total yield of the orchard?

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello BAM!,

Let's the $Y$ be the total yield of the orchard, and $T$ be the number of trees. The total yield, in peaches, is the product of the number of trees and the average yield $A$ per tree:

$$Y(T)=T\cdot A(T)$$

We are told that when there are 61 trees, the average yield is 56 peaches per tree, and that for each additional 4 trees planted, the average yield decreases by 9 peaches.

From this, we know the average yield is a linear function, contains the point $(61,56)$ and has slope $$m=\frac{\Delta A}{\Delta T}=\frac{-9}{4}=-\frac{9}{4}$$. Thus, using the point-slope formula, we have:

$$A(T)-56=-\frac{9}{4}(T-61)$$

$$A(T)=-\frac{9}{4}T+\frac{773}{4}=\frac{773-9T}{4}$$

Hence, we may now write:

$$Y(T)=\frac{T(773-9T)}{4}=\frac{1}{4}\left(773T-9T^2 \right)$$

Differentiating the yield function with respect to $T$, and equating to zero, we obtain:

$$Y'(T)=\frac{1}{4}\left(773-18T \right)=0$$

Thus, the critical value is:

$$T=\frac{773}{18}$$

We can easily see that the second derivative of the yield function is a negative constant, indicating that this critical value occurs at the global maximum.

Because the number of trees is discrete rather than continuous, we need to round this value to the nearest natural number:

$$T\approx43$$

So, in order to maximize the yield of the orchard, there needs to be 43 trees planted instead of 61. At the current number of trees the yield is:

$$Y(61)=3416$$

With 43 trees, the yield is (rounded to the nearest peach):

$$Y(43)\approx4150$$