Chapter 8.2, Problem 8P

To determine

To calculate: Constant of variation and general formula for the volume of cone.

Answer to Problem 8P

The constant of variation is $\frac{\pi }{3}$ .

A general formula for the volume of cone is:

  $V=\frac{1}{3}\pi r^{2}h$

Explanation of Solution

Given information:

The following variation:

The volume of a cone varies jointly as the height and the square of the radius of the base.

When height is $8\text{ cm}$ and base diameter is $\text{9 cm}$ , volume is $\text{54}\pi$ .

Calculation:

Let V be the volume of the cone, h be the height and r be the radius of the base. Then, V varies jointly as h and $r^2$ ,

  $V=kh{r}^2$

where k is a constant.

  $r=\frac{\text{diameter}}{2}=\frac{9}{2}=4.5\text{ cm}$ .

Substituting $V=54\pi$ , $h=8\text{ cm}$ and $r=4.5\text{ cm}$ ,

  $\begin{array}{l}V=kh{r}^2[\text{Equation]}\\ 54\pi =k\left(8\right){\left(4.5\right)}^2[\text{Substitute]}\\ 54\pi =k\left(8\right)\left(20.25\right)[\text{Find square]}\\ 54\pi =162k[\text{Multiply]}\\ k=\frac{54\pi }{162}[\text{Divide both sides by 162]}\\ k=\frac{\pi }{3}[\text{Simplify]}\end{array}$

Therefore, an equation of the given variation is $V=\frac{1}{3}\pi r^{2}h$