The diagram shows two similar solids, A and B. The ratio of the volume of solid A to the volume of solid B is 1 : 8. The total surface area of solid B is 150 cm$^2$. Find the total surface area of solid A. Answer: ___ cm$^2$

1. Find the ratio of corresponding lengths of solid A to solid B.

Let the ratio of corresponding lengths be $l_A : l_B$.
Since the ratio of volumes is $V_A : V_B = 1 : 8$,
$\left(\frac{l_A}{l_B}\right)^3 = \frac{1}{8}$
$\frac{l_A}{l_B} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}$

2. Find the ratio of surface areas and calculate the surface area of solid A.

The ratio of surface areas is $A_A : A_B = \left(\frac{l_A}{l_B}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.
Given $A_B = 150 \text{ cm}^2$,
$A_A = \frac{1}{4} \times 150 = 37.5 \text{ cm}^2$