Question

the volume of a right cone is $252\pi$ units$^3$. if its height is 21 units, find its circumference in terms of $\pi$.

Explanation:

Step1: Recall cone volume formula

The volume $V$ of a right cone is $V = \frac{1}{3}\pi r^2 h$, where $r$ is radius, $h$ is height.

Step2: Substitute known values

Substitute $V=252\pi$, $h=21$:
$$252\pi = \frac{1}{3}\pi r^2 (21)$$

Step3: Simplify to solve for $r^2$

Cancel $\pi$, calculate $\frac{1}{3} \times 21 = 7$:
$$252 = 7r^2$$
$$r^2 = \frac{252}{7} = 36$$

Step4: Solve for radius $r$

Take square root of 36:
$$r = \sqrt{36} = 6$$

Step5: Calculate circumference

Circumference $C = 2\pi r$, substitute $r=6$:
$$C = 2\pi(6) = 12\pi$$

Answer:

$12\pi$ units