# 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?

Dec 19, 2016

The ratio of their radii is $3 : 1$

#### Explanation:

Volume of bigger sphere is$\frac{4 \pi}{3} \cdot {R}^{3} = 729$ cubic in.

Volume of smaller sphere is$\frac{4 \pi}{3} \cdot {r}^{3} = 27$ cubic in. Where $R \mathmr{and} r$ are thrir respective radius

$\therefore \frac{\cancel{\frac{4 \pi}{3}} \cdot {R}^{3}}{\cancel{\frac{4 \pi}{3}} \cdot {r}^{3}} = \frac{729}{27} \mathmr{and} {\left(\frac{R}{r}\right)}^{3} = 27 \mathmr{and} {\left(\frac{R}{r}\right)}^{3} = {3}^{3} \mathmr{and} \frac{R}{r} = 3 \therefore R : r = 3 : 1$

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

Dec 21, 2016

The ratio of their radii=$3 : 1$

#### Explanation:

Vol. of a sphere=$\frac{4}{3} \pi {r}^{3}$
$\frac{4}{3} \pi {r}^{3} = 729$
$\frac{4}{3} \pi {r}^{3} = 27$
multiply both sides by$\frac{1}{\frac{4}{3} \pi}$
r^3=(27)/(4/3pi
${r}^{3} = \frac{27}{4.188790205}$
${r}^{3} = 6.445775195$
$r = \sqrt[3]{6.445775195}$
$r = 1.861$
r=±2
$\frac{4}{3} \pi {r}^{3} = 729$
multiply both sides by$\frac{1}{\frac{4}{3} \pi}$
r^3=(729)/(4/3pi
${r}^{3} = \frac{729}{4.188790205}$
${r}^{3} = 174.0359303$
$r = \sqrt[3]{174.0359303}$
$r = 5.583$ or
r=±6
$= \frac{2}{6} = 2 : 6$ 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 $3 : 1$

Dec 23, 2016

$R : r = 3 : 1$

#### Explanation:

All spheres are similar figures.

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

${\left(\frac{R}{r}\right)}^{3} = \frac{729}{27} = {9}^{3} / {3}^{3}$

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

Find the cube root of both sides.

$\frac{R}{r} = \sqrt[3]{\frac{729}{27}}$

$\frac{R}{r} = \frac{9}{3} = \frac{3}{1}$

The ratio $R : r = 3 : 1$

Ratios are always given in the simplest form.