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

The formula for volume of a sphere is:

`V=4/3pir^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:

`2304pi=4/3pir^3`

We can cancel `pi` from each side.

`2304=4/3r^3`

Multiply both sides by `3/4`.

`2304xx3/4=3/4xx4/3r^3`

`2304xx3/4=(cancel3)/(cancel4)xx(cancel4)/(cancel3)r^3`

`2304xx3/4=r^3`

`576cancel2304xx3/(cancel4)=r^3`

`576xx3=r^3`

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

`3*3*3*4*4*4=r^3`

`(3*4)(3*4)(3*4)=r^3`

`12xx12xx12=r^3`

`12=r`

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

`2xx12=24`

Vol sphere= `4/3pir^3`

`:.4/3color(red)pir^3=2304color(red)pi`

divide L.H.S and R.H.S by `color(red)pi`

`:.4/3r^3=2304`

divide L.H.S and R.H.S by `color(red)4/color(red)3`

`:.(4/3r^3)/color(red)(4/3)=2304/color(red)(4/3)`

`:.cancel4^1/cancel3^1r^3 xx cancel 3^1/cancel4^1=cancel2304^576/1 xx 3/cancel4^1`

`:.r^3=576 xx 3`

`:.r^3=1728`

`:.r=root3(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`

`:.r=3 xx 4`

`:.r=12`

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