Question

two similar pyramids have base areas of 12.2 cm² and 16 cm². the surface area of the larger pyramid is 56 cm². what is the surface area of the smaller pyramid? 40.1 cm² 42.7 cm² 52.2 cm² 59.8 cm²

Explanation:

Step1: Find the ratio of the base - areas

For two similar solids, if the ratio of the areas of corresponding bases (or any corresponding similar - shaped cross - sections) is equal to the square of the ratio of their corresponding linear dimensions. Let the base area of the smaller pyramid be $A_1 = 12.2\ cm^2$ and the base area of the larger pyramid be $A_2=16\ cm^2$. The ratio of the base areas is $\frac{A_1}{A_2}=\frac{12.2}{16}=\frac{61}{80}$.

Step2: Find the ratio of the surface areas

Since the ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding linear dimensions (which is the same as the ratio of the areas of corresponding bases), let the surface area of the smaller pyramid be $S_1$ and the surface area of the larger pyramid be $S_2 = 56\ cm^2$. We know that $\frac{S_1}{S_2}=\frac{A_1}{A_2}$.

Step3: Solve for the surface area of the smaller pyramid

We can rewrite the equation as $S_1=\frac{A_1}{A_2}\times S_2$. Substitute $A_1 = 12.2$, $A_2 = 16$, and $S_2 = 56$ into the equation: $S_1=\frac{12.2}{16}\times56=\frac{12.2\times56}{16}=\frac{683.2}{16}=42.7\ cm^2$.

Answer:

$42.7\ cm^2$