Chapter 12.5, Problem 24STP

To determine

The surface area ratio between the original and larger rectangular prism.

Answer to Problem 24STP

The surface area ratio between the original and larger rectangular prism is $\frac{1}{9}$ .

Explanation of Solution

The surface area of the rectangular prism is calculated by using the following expression:

  $A=2\left(wl+hl+hw\right)$.......(1)

Where,

Width of rectangular prism: $w$

Length of rectangular prism: $l$

Height of rectangular prism: $h$

Now, the sides of rectangular prism gets tripled then the surface area of this new or larger rectangular prism is given by:

  $A_1=2\left(w_1{l}_1+h_{1}l+h_1{w}_1\right)$.......(2)

Where,

Width of larger rectangular prism: $w_1=3w$

Length of larger rectangular prism: $l_1=3l$

Height oflarger rectangular prism: $h_1=3h$

Now, take the ratio of eq. (1) and (2):

  $\begin{array}{l}\frac{A}{A_1}=\frac{2\left(wl+hl+hw\right)}{2\left(w_1{l}_1+h_{1}l+h_1{w}_1\right)}\\ \frac{A}{A_1}=\frac{\left(wl+hl+hw\right)}{\left(3w\times 3l+3h\times 3l+3h\times 3w\right)}\\ \frac{A}{A_1}=\frac{\left(wl+hl+hw\right)}{9\left(wl+hl+hw\right)}\\ \frac{A}{A_1}=\frac{1}{9}\end{array}$

Conclusion:

Hence, the ratio is derived as $\frac{A}{A_1}=\frac{1}{9}$ for the surface area of the original and larger rectangular prism.