Chapter 3.3, Problem 51E

To determine

To find: The current instantaneous rate of increase4 of the total annual production of the apples.

Answer to Problem 51E

The current rate of the increase of the annual production of the apple is 390.

Explanation of Solution

Given Data:

The apple farmer that currently has the 156 trees yielding the average of 12 bushels of the apple per tree.

The farm is expanding farm at the rate of 13 trees per year.

The annual average yield is 1.5 bushels per tree.

Calculation:

Consider the function for the number of tress $x$ years from now is $t\left(x\right)$ .

Consider the yield per tree is $x$ years from now is $y\left(x\right)$ .

The required equation is,

  $\begin{array}{l}t\left(0\right)=156\\ y\left(0\right)=12\end{array}$

Then,

  $\begin{array}{l}{t}^{\prime }\left(0\right)=13\\ y^{\prime }\left(0\right)=1.5\end{array}$

Consider the rate of increase of the production is,

  $\begin{array}{c}\frac{d\left(ty\right)}{dx}=t\frac{dy}{dx}+y\frac{dt}{dx}\\ =t\left(0\right)y^{\prime }\left(0\right)+y\left(0\right)t^{\prime }\left(0\right)\end{array}$

Then,

  $\begin{array}{c}t\left(0\right)y^{\prime }\left(0\right)+y\left(0\right)t^{\prime }\left(0\right)=156\left(15\right)+12\left(13\right)\\ =390\end{array}$

Thus, the current rate of the increase of the annual production of the apple is 390.