Question

Find the dimensions of a rectangularprism with a volume of 162 cubic m ifthe width is one-half the length and theheight is three times the width.

Answer 1

By definition, you know that the volume of a rectangular prism is given by the formula

`\begin{gathered} V=l\cdot w\cdot h \\ \text{ Where} \\ l=\text{length} \\ w=\text{width} \\ h=\text{height} \end{gathered}`

So, you can make a drawing to understand the procedure easier

Then you have

`\begin{gathered} V=l\cdot w\cdot h \\ 162=l\cdot(l)/(2)\cdot3w \\ 162=l\cdot(l)/(2)\cdot3((l)/(2)) \\ 162=l\cdot(l)/(2)\cdot(3l)/(2) \\ 162=(3l^3)/(4) \\ \text{ Multiply both sides of the equation by }4 \\ 162\cdot4=(3l^3)/(4)\cdot4 \\ 648=3l^3 \\ \text{ Divide both sides of the equation by 3} \\ (648)/(3)=(3l^3)/(3) \\ 216=l^3 \\ \text{ Apply cube root to both sides of the equation} \\ \sqrt[3]{216}=\sqrt[3]{l^3} \\ 6=l \end{gathered}`

And so, for the other dimensions, you get

`\begin{gathered} w=(l)/(2) \\ w=(6)/(2) \\ w=3 \end{gathered}`
`\begin{gathered} h=3\cdot w \\ h=3\cdot3 \\ h=9 \end{gathered}`

Therefore, the dimensions of the rectangular prism are 6 meters in length, 3 meters in width, and 9 meters height.