Question

The volumes of two spheres are `729` `i n^3` and `27` `i n^3`. What is the ratio of their radii, rounded to the nearest whole number?

Answer 1

The ratio of their radii is `3:1`

Explanation:

Volume of bigger sphere is`(4pi)/3*R^3 = 729` cubic in.

Volume of smaller sphere is`(4pi)/3*r^3 = 27` cubic in. Where `R and r` are thrir respective radius

`:.(cancel((4pi)/3)*R^3) /(cancel((4pi)/3)*r^3)=729/27 or (R/r)^3=27 or (R/r)^3=3^3 or R/r=3 :.R:r =3:1`

The ratio of their radii is `3:1` [Ans]

Answer 2

The ratio of their radii=`3 : 1`

Explanation:

Vol. of a sphere=`4/3pir^3`
`4/3pir^3=729`
`4/3pir^3=27`
multiply both sides by`(1)/(4/3pi)`
`r^3=(27)/(4/3pi`
`r^3=27/4.188790205`
`r^3=6.445775195`
`r=root3(6.445775195)`
`r=1.861`
`r=±2`
`4/3pir^3=729`
multiply both sides by`(1)/(4/3pi)`
`r^3=(729)/(4/3pi`
`r^3=729/4.188790205`
`r^3=174.0359303`
`r=root3(174.0359303)`
`r=5.583` or
`r=±6`
`=2/6=2:6` ratio of their radii

Ratios are always in simplest form, so `1 : 3`

However we are given the ratios as
Big sphere : small sphere, so the ratio is `3 : 1`

Answer 3

`R : r = 3:1`

Explanation:

All spheres are similar figures.

Their volumes are in the same ratio as the cubes of their radii.

`(R/r)^3 = 729/27 = 9^3/3^3`

Note that we are not asked to find the actual lengths of their radii, only the ratio.

Find the cube root of both sides.

`R/r = root3(729/27)`

`R/r = 9/3 = 3/1`

The ratio `R : r = 3:1`

Ratios are always given in the simplest form.