Chapter 8.1, Problem 19P

To determine

To find the constant of proportionality in terms of $\pi$ and write the formula for the volume of sphere in terms of diameter.

Answer to Problem 19P

The constant of proportionality is $\frac{\pi }{6}$ and the formula for volume of the sphere is $v=\frac{\pi }{6}{\text{ d}}^{\text{3}}$ .

Explanation of Solution

Given information:

The given statement is:

“The volume of sphere is directly proportional to the cube of its diameter and is $288\pi {\text{ cm}}^{\text{3}}$ when the diameter is $\text{12 cm}$ . Find the constant of proportionality in terms of $\pi$ . Then write the formula for the volume of sphere in terms of diameter.”

Calculation :

Let the volume be $v$ and the diameter be $d$ .

The constant of proportionality can be written as:

  $\begin{array}{l}\frac{v}{d^3}=\frac{288\pi }{{12}^3}\\ \frac{v}{d^3}=\frac{\pi }{6}\\ v=\frac{\pi }{6}{d}^3\end{array}$

Hence, the constant of proportionality is $\frac{\pi }{6}$ and the formula for volume of the sphere is $v=\frac{\pi }{6}{\text{ d}}^{\text{3}}$ .