To solve this problem, we should understand the relationship between scale factors for volume, area, and linear measurements of similar shapes. In similarity, corresponding lengths of similar shapes are proportional.
If the volume scale factor is the cube of the linear scale factor, then to find the linear scale factor when given the volume scale factor, we take the cube root of the volume scale factor.
(a) To find the linear scale factor:
Given the volume scale factor is 27, the linear scale factor (let's call it 'L') is the cube root of 27.
\[ L = \sqrt[3]{27} \]
\[ L = 3 \]
This means the corresponding lengths of the two cylinders are in a ratio of 1:3.
(b) To find the area scale factor:
The area scale factor is the square of the linear scale factor. Since we found that the linear scale factor is 3, we square this value to find the area scale factor (let's call it 'A').
\[ A = L^2 \]
\[ A = 3^2 \]
\[ A = 9 \]
Therefore, the area scale factor of the two similar cylinders is 9. This means that the ratio of the areas of corresponding sections of the cylinders is 1:9.
