Chapter 4.7, Problem 33E

To determine

To find: the function for $W$ that gives the volume of cone and interpret the function $W\left(3\right)$ .

Answer to Problem 33E

  $W\left(3\right)=486\pi .$

Explanation of Solution

Given:

  

Concept used:

The volume of cone is $\frac{1}{3}\pi r^{2}h$ .

A composite function is a function that depend on another function. A composite function is created when one function is substituted into another function.

for example $f\left(g\left(x\right)\right)$ is the composite function that is formed when $g\left(x\right)$ is substituted for $x$ in $f\left(x\right)$ .

The domain of $f$ becomes $g\left(x\right)$ .

In this case the value $g\left(x\right)$ is substitute in the domain of the $f\left(x\right)$ .

Calculation:

The volume of cone is $\frac{1}{3}\pi r^{2}h$ .

  $\begin{array}{l}V\left(x\right)=\frac{1}{3}\pi {\left(3x\right)}^2\left(x+3\right)\text{yd}\text{.}\\ V\left(x\right)=3\pi x^2\left(x+3\right)\text{yd}.\end{array}$

Since, $1\text{ yard=}3\text{ feet }$ .

According to the given $W$ is the function that gives the volume of cone when $x$ is measure in feet.

  $\begin{array}{l}W\left(3\right)=3\pi {\left(3\right)}^2\left(3+3\right)\left(3\right)\text{feet}.\\ \Rightarrow 486\pi .\end{array}$

Hence, $W\left(3\right)=486\pi .$