Question

a rectangular prism has a volume of 144 cubic cm. the length of the prism is 3 times the width. The height is 2 times as much as the length. what are the dimensions of the prism?

Answer 1

Given:

Volume of prism is, V = 144 cubic cm.

The length of the prism is 3 times the width.

The height is 2 times as much as the length.

The objective is to find the dimensions of the prism.

Consider the width of the prism as x.

Consider the length as l, which is 3 times the width of the prism.

`l=3x`

Consider the height as h, which is 2 times as much as the length.

`\begin{gathered} h=2l \\ =2(3x) \\ =6x \end{gathered}`

Since, the formula of volume rectangular prism is,

`V=l\cdot b\cdot h`

Now, substitute the obtained values in the above equation.

`\begin{gathered} 144=3x\cdot x\cdot6x \\ 144=18x^3 \\ x^3=(144)/(18) \\ x^3=8 \\ x=\sqrt[3]{8} \\ x=2 \end{gathered}`

Thus, the width of the prism is, b = 2 cm.

The length of the prism will be,

`\begin{gathered} l=3x \\ =3(2) \\ =6\text{ cm} \end{gathered}`

Similalry, the height of the prism will be,

`\begin{gathered} h=6x \\ =6(2) \\ =12\text{ cm} \end{gathered}`

Hence, the length is 6 cm, width is 2 cm and the height is 12 cm.