Chapter 1.7, Problem 4QR

Solution

The solution of the formula $V=\frac{1}{3}\pi r^{2}h$ for h .

The solution of the formula $V=\frac{1}{3}\pi r^{2}h$ for h is $h=\frac{3V}{\pi r^2}$ .

Given:

The formula of volume of right circular cone, $V=\frac{1}{3}\pi r^{2}h$ .

Concept Used:

Applying the same operation (addition, subtraction, multiplication and division) on both sides of an equation doesn’t affect the equality of the equation.

The application of operation can be used to isolate a variable or solve an equation of a variable.

Calculation:

Consider the formula, $V=\frac{1}{3}\pi r^{2}h$ .

To solve the equation for h , isolate h as follows,

  $\begin{array}{c}V=\frac{1}{3}\pi r^{2}h\text{ Multiply both sides by }3\\ 3\times V=3\times \frac{1}{3}\pi r^{2}h\text{ Simplify}\\ 3V=\pi r^{2}h\text{ Divide both sides by }\pi \text{ or multiply by }\frac{1}{\pi }\\ \frac{1}{\pi }\times 3V=\frac{1}{\pi }\times \pi r^{2}h\text{ Simplify}\end{array}$

  $\begin{array}{c}\frac{3V}{\pi }=r^{2}h\text{ Divide both sides by }{r}^2\\ \frac{3V}{\pi r^2}=\frac{r^{2}h}{r^2}\text{ Simplify}\\ \frac{3V}{\pi r^2}=h\text{ Rewirte}\\ h=\frac{3V}{\pi r^2}\end{array}$

Hence the required solution is $h=\frac{3V}{\pi r^2}$ .

Conclusion:

The solution of the formula $V=\frac{1}{3}\pi r^{2}h$ for h is $h=\frac{3V}{\pi r^2}$ .