# If the volume of a sphere is 2304pi, then what is the diameter?

## My teacher requires work to be shown. If you can show work, if it's not too much trouble, please do!!! thank you!!!

Mar 3, 2017

The diameter is $24$.

#### Explanation:

The formula for volume of a sphere is:

$V = \frac{4}{3} \pi {r}^{3}$

Since we know the volume, and have to find the diameter (twice the radius), we need to determine the radius first.

Hence, using the given data:

$2304 \pi = \frac{4}{3} \pi {r}^{3}$

We can cancel $\pi$ from each side.

$2304 = \frac{4}{3} {r}^{3}$

Multiply both sides by $\frac{3}{4}$.

$2304 \times \frac{3}{4} = \frac{3}{4} \times \frac{4}{3} {r}^{3}$

$2304 \times \frac{3}{4} = \frac{\cancel{3}}{\cancel{4}} \times \frac{\cancel{4}}{\cancel{3}} {r}^{3}$

$2304 \times \frac{3}{4} = {r}^{3}$

$576 \cancel{2304} \times \frac{3}{\cancel{4}} = {r}^{3}$

$576 \times 3 = {r}^{3}$

To find the cube root, we separate the factors of the number on the left.

$3 \cdot 3 \cdot 3 \cdot 4 \cdot 4 \cdot 4 = {r}^{3}$

$\left(3 \cdot 4\right) \left(3 \cdot 4\right) \left(3 \cdot 4\right) = {r}^{3}$

$12 \times 12 \times 12 = {r}^{3}$

$12 = r$

Since the radius ($r$) is $12$, the diameter, which is $2 r$, will be:

$2 \times 12 = 24$

Mar 3, 2017

$24$

#### Explanation:

Vol sphere= $\frac{4}{3} \pi {r}^{3}$

$\therefore \frac{4}{3} \textcolor{red}{\pi} {r}^{3} = 2304 \textcolor{red}{\pi}$

divide L.H.S and R.H.S by $\textcolor{red}{\pi}$

$\therefore \frac{4}{3} {r}^{3} = 2304$

divide L.H.S and R.H.S by $\frac{\textcolor{red}{4}}{\textcolor{red}{3}}$

$\therefore \frac{\frac{4}{3} {r}^{3}}{\textcolor{red}{\frac{4}{3}}} = \frac{2304}{\textcolor{red}{\frac{4}{3}}}$

$\therefore {\cancel{4}}^{1} / {\cancel{3}}^{1} {r}^{3} \times {\cancel{3}}^{1} / {\cancel{4}}^{1} = {\cancel{2304}}^{576} / 1 \times \frac{3}{\cancel{4}} ^ 1$

$\therefore {r}^{3} = 576 \times 3$

$\therefore {r}^{3} = 1728$

$\therefore r = \sqrt[3]{1728}$

:.r=root3(3*3*3*4*4*4

:. color(red)(root3 (3) xx color(red)( root 3color(red)(3) xx root 3 (3) = 3
:.color(red)(root 3(4) xx root 3(4) xx root 3 (4) = 4

$\therefore r = 3 \times 4$

$\therefore r = 12$

:.color(red)(Diametercolor(red)= 12 xx 2=24