Question

Find the volume of a right circular cone that has a height of 15 m and a base with a circumference of 18.5 m. Round your answer to the nearest tenth of a cubic meter.

Answer 1

In this problem, we asked to find the volume of a cone. The formula for this is:

`\frac{\pi {r}^(2) h }{3}`
To solve that equation, we need to know height and radius. We know height, so let's find the radius.

They gave you the circumference and we can use it.

The formula for the circumference is:

`2\pi r = c`
Plug in the givens:

`2 * 3 r = 18.5 \\ \\ 6r = 18.5 \\ \\ r = 3.1`
Now, plug the values in the first formula:

`\frac{3 * {3.1}^(2) * 15}{3} = {3.1}^(2) * 15 = 9.61 * 15 = 144.15`
So the volume is 144.15 m3.

(I assumed π = 3.)

Answer 2

I hate the rounding. Why ruin a nice exact answer with an approximation?

I like doing the general solution first and plugging in the numbers last.

The usual formula for the volume of a cone is

`V = \frac 1 3 \pi r^2 h`

We're given circumference

`C = 2 \pi r`

so

`r = (C)/(2\pi)`

`V = \frac 1 3 \pi (C/(2\pi))^2 h = ( hC^2)/(12 \pi)`

Substituting,

`V = (15 (18.5)^2)/(12 \pi) = (6845)/(16\pi) \approx 136.2 \textrm{ m}^3`