Question

Find the volume of a right circular cone that has a height of 12.1 m and a base with a circumference of 17.7 m. Round your answer to the nearest tenth of a cubic meter.

Answer 1

`V_(cone)=100.6 m^(3)`

Explanation:

The volume of a cone is defined as:

`V_(cone)=(1)/(3)A_(base)h`

So, a cone has a circular base, and its area is:

`A_(base)=\pi r^(2)`

Replacing this relation in the volume equation:

`V_(cone)=(1)/(3)\pi r^(2)h`

Now, the problem gives us the height but no the radius, instead it gives the circumference length, which we can use to find the radius:

`L=2\pi r\\r=(L)/(2 \pi)\\ r=(17.7m)/(2 \pi)\\r=(8.85)/(\pi)m`

Then, we use this radius to find the volume:

`V_(cone)=(1)/(3)\pi ((8.85)/(\pi)m)^(2)12.1m`

`V_(cone)=(315.9)/(\pi) m^(3)`

Using
`\pi \approx 3.14`:

`V_(cone)=(315.9)/(3.14) m^(3)=100.6 m^(3)`

Therefore, the volume is
`V_(cone)=100.6 m^(3)`

Answer 2

The volume of a right circular cone is

`V=(1)/(3)\pi r^2\cdot h`.

If a circumference of a base circle is 17.7 m, then using formula
`l=2\pi r` for circumference of a circle, you can find the radius:

`17.7=2\pi r,\\ \\ r=(17.7)/(2\pi) =(8.85)/(\pi)`.

Then the volume is:

`V=(1)/(3)\pi r^2\cdot h =(1)/(3)\pi \left((8.85)/(\pi)\right)^2\cdot 12.1=(315.90075)/(\pi)`.