Chapter 2.3, Problem 49E

To determine

The reason behind the rate of change with respect to radius is circumference.

Explanation of Solution

Given information:

The number of members presently $65$ , rent is $\$250$ bushels and increase in number of members be 6 members and increase in rent is $\$10$ bushels.

Concept used:

Use the product rule $\frac{d}{dx}\left(u.v\right)=u.\frac{dv}{dx}+v.\frac{du}{dx}$ .

Calculation:

Let the number of members be represented by function be $f\left(x\right)$ and the rent be represented by the function $g\left(x\right)$ .

So the number of members in previous year is $f\left(x\right)=65$ .

The rent is $g\left(x\right)=250$ .

Increase in number of trees is $f^{\prime }\left(x\right)=6$ .

Increase in production is $g^{\prime }\left(x\right)=10$ .

Substitute the values in the product rule,

  $\begin{array}{c}\frac{d}{dx}\left(f\left(x\right).g\left(x\right)\right)=f\left(x\right).g^{\prime }\left(x\right)+g\left(x\right).f^{\prime }\left(x\right)\\ =65\times 10+250\times 6\\ =650+1500\\ =2150\end{array}$

Conclusion: The rate of increase in total production is $2150$ .