# Geometry help? Volume of a cone.

## The volume of a cone is 1014𝜋 𝑖𝑛^3. The height of the cone is 18 𝑖𝑛. What is the exact circumference of the base of the cone? Do NOT round at any point!

Feb 27, 2018

$\text{circumference "=26pi" inches}$

#### Explanation:

$\text{to find circumference we require to know the radius r}$

$\text{using the following formulae}$

•color(white)(x)V_(color(red)"cone")=1/3pir^2hlarrcolor(blue)"volume of cone"

• " circumference (C)"=2pir

${V}_{\textcolor{red}{\text{cone}}} = \frac{1}{3} \pi {r}^{2} \times 18 = 6 \pi {r}^{2}$

$\text{now volume is given as } 1014 \pi$

$\Rightarrow 6 \pi {r}^{2} = 1014 \pi$

$\text{divide both sides by } 6 \pi$

(cancel(6pi) r^2)/cancel(6pi)=(1014cancel(pi))/(6cancel(pi)

$\Rightarrow {r}^{2} = \frac{1014}{6} = 169$

$\Rightarrow r = \sqrt{169} = 13$

$\Rightarrow C = 2 \pi \times 13 = 26 \pi \leftarrow \textcolor{red}{\text{exact value}}$

Feb 27, 2018

The volume of a cone is $V = \frac{\setminus \pi {r}^{2} h}{3}$

#### Explanation:

$1014 \setminus \pi = \frac{\setminus \pi {r}^{2} \cdot 18}{3}$

The $\setminus \pi$ on each side of the equal sign will cancel, so

$1014 = \frac{{r}^{2} \cdot 18}{3}$

Multiply both sides by 3

$3042 = {r}^{2} \cdot 18$

Then divide both sides by 18

$169 = {r}^{2}$

Then, take the square root of both sides

$\sqrt{169} = {\sqrt{r}}^{2}$

$\pm 13 = r$

Since this is a distance , use the positive square root since distances can't be negative, so r = 13.

Then, the circumference of a circle is $2 \setminus \pi r$

So, $2 \cdot 13 \setminus \pi \to 26 \setminus \pi$

That is your answer and is an exact value since it is in terms of $\setminus \pi$