Question

Solve the equation for the indicated variable.

$V=\frac{1}{3}\pi r^{2}h;\text{\hspace{0.33em}for\hspace{0.33em}}r$

Short answer

The required value is $r=\sqrt{\frac{3V}{\pi h}}$.

Step-by-step solution

Step 1. Definition of equation and properties of equality.

Equation is a mathematical statement in which two expressions are equal to each other.

According to the multiplication property of equality if $a=b$, then $ac=bc$.

According to the division property of equality if $a=b$, then $\frac{a}{c}=\frac{b}{c}$, where $c\ne 0$.

Step 2. Given Information.

The given equation is:

$V=\frac{1}{3}\pi r^{2}h$

Step 3. Solve the given equation for required variable.

Using the multiplication property of equality, multiply both sides by$3$ to simplify the given equation.

$$\begin{gathered} 3\times V=3\times \frac{1}{3}\pi r^{2}h \\ 3V=\pi r^{2}h \end{gathered}$$

Using the division property of equality, divide both sides by $\pi h$.

$$\begin{gathered} \frac{3V}{\pi h}=\frac{\pi r^{2}h}{\pi h} \\ \frac{3V}{\pi h}=r^2 \end{gathered}$$

Take square root on both sides

$\pm \sqrt{\frac{3V}{\pi h}}=r$

In the given equation represents the volume of a cone and$r$ is the radius of the cone. Since the radius is always a non-negative value, therefore the value of$r$ cannot be a negative value.

Thus, the required value is $r=\sqrt{\frac{3V}{\pi h}}$.