Chapter 12.8, Problem 28STP

To determine

The correct option of the volume ratio smaller prism to larger prism is to be selected from the following four options:

  $8:27$

  $2:3$

  $4:9$

  $1:2$

Answer to Problem 28STP

The correct option is A. The value of the volume ratio smaller prism to larger prism $8:27$ .

Explanation of Solution

Given:

  

  $\begin{array}{l}{V}_1=2250c{m}^3\\ V_2=18c{m}^3\\ L_1=45cm\\ L_2=xcm\end{array}$

Concept Used:

The volume of the rectangular prism is given by the following expression:

  $V=whl$

Where,

Width of rectangular prism: $w$

Height of rectangular prism: $h$

Length of rectangular prism: $l$

Calculation:

The volume of the larger rectangular prism is calculated as:

  $\begin{array}{l}{V}_1=w_1{h}_1{l}_1\\ V_1=\left(3\right)\left(21\right)\left(12\right)f{t}^3\end{array}$

The volume of the smaller rectangular prism is calculated as:

  $\begin{array}{l}{V}_2=w_2{h}_2{l}_2\\ V_2=\left(2\right)\left(14\right)\left(8\right)f{t}^3\end{array}$

The value of the volume ratio smaller prism to larger prism:

  $\begin{array}{l}\frac{V_2}{V_1}=\frac{\left(2\right)\left(14\right)\left(8\right)f{t}^3}{\left(3\right)\left(21\right)\left(12\right)f{t}^3}\\ \frac{V_2}{V_1}=\frac{\left(14\right)\left(8\right)}{\left(3\right)\left(21\right)\left(6\right)}\\ \frac{V_2}{V_1}=\frac{8}{27}\\ V_2:V_1=8:27\end{array}$

Conclusion:

Hence, the value of the volume ratio smaller prism to larger prism is calculated as $V_2:V_1=8:27$ .