Answer to Question #168646 in Geometry for Adam

Question

Solid A and solid B are similar. The surface area of solid A is 675 m² and the surface area of solid B is 432 m². If the volume of solid B is 960 m³, find the volume of solid A.

Answer 1

Answer

• As the shape of solids are not given, lets start from the definition of the volume: $\small V=L\times W\times H$

$\qquad\qquad \begin{aligned} \small V_b&= \small L_b\times W_b\times H_b\\ \small V_a&= \small L_a\times W_a\times H_a \end{aligned}$

• Since the solids are similar,

$\qquad\qquad \begin{aligned} \small \frac{Area_a}{Area_b}&= \small (\frac{L_a}{L_b})^2=(\frac{W_a}{W_b})^2=(\frac{H_a}{H_b})^2\\ \small \frac{675}{432}=(\frac{5}{4})^2&= \small \small (\frac{L_a}{L_b})^2=(\frac{W_a}{W_b})^2=(\frac{H_a}{H_b})^2\\ \\ Then,\\ \\ \small L_a&= \small \frac{5L_b}{4}\\ \small W_a&= \small \frac{5W_b}{4}\\ \small H_a&= \small \frac{5H_b}{4}\\ \\ \therefore\small V_a&= \small \frac{5L_b}{4}\times \frac{5W_b}{4}\times \frac{5H_b}{4}\\ &= \small (\frac{5}{4})^3\times V_b\\ &= \small 1875m^3 \end{aligned}$