Chapter 4.7, Problem 33E

To determine

To write: a function $V$ for the volume of the circular cone and to write a function $W$ that gives the volume of the cone when $x$ is measured in feet and to find the interpret $W\left(3\right)$ .

Answer to Problem 33E

The volume of the circle cone is $V=3\pi r^2\left(x+3\right)$ .

The function is $W\left(x\right)=\frac{\pi }{3}{x}^2\left(\frac{1}{3}x+3\right)$ .

The volume of circular cone is $12\pi$ cubic yards.

Explanation of Solution

Given information:

  

Calculation:

The volume of circular cone is

  $\begin{array}{c}V=\frac{1}{3}\pi {\left(radius\right)}^2\left(height\right)\\ =\frac{1}{3}\cdot \pi {\left(3x\right)}^2\left(x+3\right)\\ =3\pi r^2\left(x+3\right)\end{array}$

Because $1\text{ft}=\frac{1}{3}\text{yd}$ , so

  $\begin{array}{c}W\left(x\right)=V\left(\frac{1}{3}x\right)\\ =3\pi {\left(\frac{1}{3}x\right)}^2\left(\frac{1}{3}x+3\right)\\ =\frac{\pi }{3}{x}^2\left(\frac{1}{3}x+3\right)\end{array}$

Therefore, the function is $W\left(x\right)=\frac{\pi }{3}{x}^2\left(\frac{1}{3}x+3\right)$ .

Now evaluate $W\left(3\right)$ by substituting $x=3$ ,

  $\begin{array}{c}W\left(3\right)=\frac{\pi }{3}{\left(3\right)}^2\left(\frac{1}{3}\cdot 3+3\right)\\ =3\pi \left(4\right)\\ =12\pi \end{array}$

Therefore, $W\left(3\right)=12\pi$ . This represents that when $x$ is 3 feet the volume of circular cone is $12\pi$ cubic yards.