Question

two similar cylinders have surface areas of 24π cm² and 54π cm². the volume of the smaller cylinder is 16π cm³. what is the volume of the larger cylinder? 36π cm³ 46π cm³ 48π cm³ 54π cm³

Explanation:

Step1: Find the ratio of the surface - areas

The ratio of the surface - areas of two similar solids is equal to the square of the ratio of their corresponding linear dimensions. Let the ratio of the linear dimensions of the smaller cylinder to the larger cylinder be $k$. Then $\frac{S_1}{S_2}=k^{2}$, where $S_1 = 24\pi$ and $S_2=54\pi$. So $k^{2}=\frac{24\pi}{54\pi}=\frac{4}{9}$, and $k = \frac{2}{3}$.

Step2: Find the ratio of the volumes

The ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding linear dimensions. Let $V_1$ be the volume of the smaller cylinder and $V_2$ be the volume of the larger cylinder. Then $\frac{V_1}{V_2}=k^{3}$. Since $k=\frac{2}{3}$, $k^{3}=\frac{8}{27}$. We know that $V_1 = 16\pi$. So $\frac{16\pi}{V_2}=\frac{8}{27}$.

Step3: Solve for $V_2$

Cross - multiply to get $8V_2=16\pi\times27$. Then $V_2=\frac{16\pi\times27}{8}=54\pi$ $cm^{3}$.

Answer:

$54\pi$ $cm^{3}$