# The volumes of two similar solids are 53 cm3 and 1113 cm3. Which is the ratio of the corresponding sides?

Jun 13, 2017

The ratio of the corresponding sides is $0.3625 : 1$

#### Explanation:

Similar solids means that all dimensions are proportional and all angles are equal or if it involves circular surfaces, their radii too are proportionals.

In such cases if the ratio of corresponding sides (or dimensions) is say $x$, then their volumes are in the ratio ${x}^{3}$. In other words, if ratio of volumes is $v$, then ratio of dimensions (corresponding sides) is $\sqrt[3]{v}$.

It is given that volumes are in the ratio $\frac{53}{1113} = \frac{53}{53 \times 21} = \frac{1}{21}$

Hence ratio of corresponding sides is $\sqrt[3]{\frac{1}{21}} = \frac{\sqrt[3]{1}}{\sqrt[3]{21}} = \frac{1}{2.759} = 0.3625$ or $0.3625 : 1$

Jun 13, 2017

$1 : \sqrt[3]{21}$

#### Explanation:

let say the $k$ is the ratio of corresponding side, where $l$ and $L$ are for the length of sides of solid respectively.

$l = k L$$\to k = \frac{l}{L}$
$l \cdot l \cdot l = k L \cdot k L \cdot k L$

${l}^{3} = {k}^{3} \cdot {L}^{3}$ where ${l}^{3} = 53 \mathmr{and} {L}^{3} = 1113$

$53 = {k}^{3} \cdot 1113$

$\frac{153}{1113} = {k}^{3}$

$\frac{1}{21} = {k}^{3}$

$\sqrt[3]{\frac{1}{21}} = k \to \frac{1}{\sqrt[3]{21}}$,