Question

The relationship between the volumes and the lengths of two cereal boxes is represented by

$\frac{volumeofboxA}{volumeofboxB}={\left(\frac{lengthofboxA}{lengthofboxB}\right)}^3$

Box $A$ has a volume of $192$ cubic inches and a length of $8$ inches. Box $B$ has a volume of $375$ cubic inches. What is the length of Box $B$? Justify your answer.

Answer 1

Find the length of the box $B$.

We have given that the Box $A$ has a volume of $192$ cubic inches and a length of $8$ inches. and Box $B$ has a volume of $375$ cubic inches.

The relationship between the volumes and the lengths of two cereal boxes is represented by

$\frac{volumeofboxA}{volumeofboxB}={\left(\frac{lengthofboxA}{lengthofboxB}\right)}^3$

Let $x$ be the length of the box $B$.

Substituting these values in the given relationship, we get

$$\begin{gathered} \frac{192}{375}={\left(\frac{8}{x}\right)}^3 \\ \Rightarrow \sqrt[3]{\frac{192}{375}}=\frac{8}{x} \\ \Rightarrow \sqrt[3]{\frac{64\times 3}{125\times 3}}=\frac{8}{x} \\ \Rightarrow \sqrt[3]{\frac{64}{125}}=\frac{8}{x} \\ \Rightarrow \frac{4}{5}=\frac{8}{x} \\ \Rightarrow x=\frac{8\times 5}{4} \\ \Rightarrow x=\frac{40}{4} \\ \Rightarrow x=10in. \end{gathered}$$

Hence, the length of the box $B$ is $10$ inches.