In a cone with a radius equal to its height, what is the radius when the volume is 72 pi units?

To determine the radius of the cone when its volume is (72\pi) cubic units and the radius is equal to its height, we can use the formula for the volume of a cone, which is given by:

[

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

]

In this case, since the radius (r) is equal to the height (h), we can replace (h) with (r) in the volume formula:

[

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

]

Setting the volume (V) equal to (72\pi), we have:

[

\frac{1}{3} \pi r^3 = 72\pi

]

We can simplify this equation by dividing both sides by (\pi):

[

\frac{1}{3} r^3 = 72

]

Next, we can eliminate the fraction by multiplying both sides by 3:

[

r^3 = 216

]

Now, to solve for (r), we take the cube root of both