these solids are similar. find the surface area of the smaller solid.\n\nimage of two similar cylinders…

these solids are similar. find the surface area of the smaller solid.\n\nimage of two similar cylinders: larger cylinder with r = 9 cm, sa = 324 cm²; smaller cylinder with r = 3 cm, sa =? cm²\n\nr = 9 cm\nsa = 324 cm²\n\nr = 3 cm\nsa =? cm²

these solids are similar. find the surface area of the smaller solid.\n\nimage of two similar cylinders: larger cylinder with r = 9 cm, sa = 324 cm²; smaller cylinder with r = 3 cm, sa =? cm²\n\nr = 9 cm\nsa = 324 cm²\n\nr = 3 cm\nsa =? cm²

Answer

Explanation:

Step1: Find similarity ratio

The ratio of radii is $\frac{3}{9} = \frac{1}{3}$.

Step2: Relate surface area ratio

For similar solids, the ratio of surface areas is the square of the similarity ratio: $\left(\frac{1}{3}\right)^2 = \frac{1}{9}$.

Step3: Calculate smaller surface area

Let $SA_{\text{small}}$ be the surface area of the smaller cylinder. Set up the proportion: $\frac{SA_{\text{small}}}{324} = \frac{1}{9}$ Solve for $SA_{\text{small}}$: $SA_{\text{small}} = 324 \times \frac{1}{9}$

Answer:

$36$ $\text{cm}^2$