Modelling the orchard yields with diminishing returns per tree

This is Exercise 7.9 in the University of Washington's precal book.

Sylvia has an apple orchard. One season, her 100 trees yielded 140 apples per tree. She wants to increase her production by adding more trees to the orchard. However, she knows that for every 10 additional trees she plants, she will lose 4 apples per tree (i.e., the yield per tree will decrease by 4 apples). How many trees should she have in the orchard to maximize her production of apples?

Instructing a precal class, I walked through my approach to answering this in class (without preparing) and ended up doing something strange that may have confused the students. How would you clearly explain a way to get the function for the number of apples that we are trying to maximize? More generally, how would approach answering this question? It'd be nice to collect a some different approaches to demonstrate to my students that there is not a single rigid correct way to solve a problem.

Solution 1:

Let $t=\text{the number of trees}$

Let $y=\text{the yield per tree}$

We know that when $t=100$, $y=140$. Furthermore, we know that for every $10$ additional trees, the yield decreases by $4$. I'm going to assume that this means that for every additional tree, the yield decreases by $0.4$ (obviously this is an average yield). If I increase the number of trees by $M$, then $t$ and $y$ change as follows:

\begin{align} t&=100+M \\ y&=140-0.4M \end{align}

We want the total yield to be as large as possible. This means that $ty$ has to be as large as possible:

\begin{align} ty&=(100+M)(140-0.4M) \\ &=14000-40M+140M-0.4M^2 \\ &=-0.4M^2+100M+14000 \\ &=-0.4(M^2-250M-35000) \end{align}

Completing the square on $M^2-250M-35000$, we have

\begin{align} M^2-250M-35000&=(M-125)^2-125^2-35000\\ &=(M-125)^2-50,625 \end{align}

Hence,

\begin{align} ty&=-0.4[(M-125)^2-50,625] \\ &=-0.4(M-125)^2+20,250 \end{align}

Since $-0.4(M-125)^2$ is always non-positive, the maximum yield is $20,250$, which occurs when $M=125$. Thus, the total number of trees in the orchard should be $225$.