Question

The fruit tree yield per tree in an orchard containing 20 trees is 252 pounds per tree each year. Due to crowding, the yield decreases by 3 pounds per tree for every additional tree planted. If you wish to maximize the total annual yield, what is the total number of trees that should be in the orchard?

Answer 1

52

Step-by-step explanation:

Let the number of fruit trees planted additionally be $n$

Initially it is given that there are $20$ trees.

Number of trees after planting $n$ additional trees is $n+20$

Let the yield due to each tree after planting $n$ additional trees be $y$

Initially it is given that $y=252$

Yield due to each tree after planting $n$ trees is $y=252-(3\times n)$

$\text{total yield}=\text{yield for each tree}\times\text{total number of trees}$

$\text{total yield}=(252-3n)(20+n)$

                                 =$252\times 20-192n-3n^{2}$

To maximise yield,we take that value of $n$ for which $\frac{d\text{total yield}}{dn}$$=0$

$\frac{d\text{total yield}}{dn}=\frac{d(252\times 20+192n-3n^{2})}{dn}$ =$192-6n$

So,$192=6n$ and $n=32$

So,32 additional trees has to be planted to maximise yield.

So,there should be 52 trees in total