Question

The yield of orange trees is reduced if they are planted too close together.

If there are $30$ trees per acre, each tree produces $400$ oranges.

For each additional tree in the acre, the yield is reduced by $7$ oranges per tree.

How many trees per acre yield the largest crop for farmer Boyles?

Answer 1

Find how many trees per acre yield the largest crop for farmer Boyles.

Let $x$ be an additional tree planted per acre.

Then in one acre, there will be $(30+x)$ trees.

Since yield per tree reduces by $10$apples after planting $x$ trees per acre.

New yield per tree per acre becomes $\left(400-10x\right)$.

Let $y$ denotes apples per acre.

Multiply $\left(30+x\right)$ by $\left(400-10x\right)$ and equate with $y$.

$$\begin{gathered} y=(30+x)(400-10x) \\ \therefore y=12000+100x-10x^2 \end{gathered}$$

The power rule of differentiation states that $\frac{d}{dx}\left(x^p\right)=p{x}^{p-1}$, where $p$ is any real number.

Use the power rule of differentiation to take its first differential with respect to variable ie $x$ and equate with $0$

$$\begin{gathered} 0=100-20x \\ \therefore x=5 \end{gathered}$$

So for optimizing the yield no of trees would be $30+5,\mathrm{i}.\mathrm{e}.35$

Hence, the trees per acre yield the largest crop for farmer Boyles is $35.$