MHB Jordan's Question from Facebook

[Sudharaka]

Jordan of Facebook writes:

I've tried everything but I keep coming up with 58 as my answer which WebWork tells me is wrong. Please help!

An orchard contains 50 peach trees with each tree yielding an average of 49 peaches. For each 10 additional trees planted, the average yield per tree decreases by 12 peaches. How many trees should be planted to maximize the total yield of the orchard?

 

[MarkFL]

Hello Jordan,

Let's let $T$ be the number of trees and $P$ be the total yield of peaches.

We are given that:

$\displaystyle \frac{\Delta P}{\delta T}=\frac{-12}{10}=-\frac{6}{5}$

We are also given the point $(T,P)=(50,49)$

Using the point-slope formula, we find:

$\displaystyle P-49=-\frac{6}{5}(T-50)$

$\displaystyle P(T)=-\frac{6}{5}T+109$

Now, the total yield $Y$ of the orchard is the number of trees times the average yield per tree:

$\displaystyle Y=T\cdot P(T)$

$\displaystyle Y(T)=T\left(-\frac{6}{5}T+109 \right)$

We know this is a parabolic yield function, and the axis of symmetry (where the vertex, or maximum) will occur midway between the roots, which are:

$\displaystyle T=0,\,\frac{545}{6}$

And so the axis of symmetry is at:

$\displaystyle T=\frac{545}{12}$

Now, since $T$ represents the number of trees, the variable $T$ is discrete rather than continuous, so we must round to the nearest integer:

$T=45$

This is the number of trees that will maximize the yield of peaches from the orchard.