Calculating Cylinder Surface Area Using Scale Factor Ratios

The linear scale factor provided for the cylinders is $3:5$ .
To calculate the surface area rather than linear dimensions, the ratio of the surface areas must be determined first.
The mathematical process for finding the surface area ratio involves squaring both numbers in the linear scale factor: - The square of $3$ is evaluated: $3^2 = 9$ . - The square of $5$ is evaluated: $5^2 = 25$ .
Therefore, the derived ratio for the surface areas is $9:25$ .

In this mathematical application, the ratio represents a proportional relationship between the two figures.
An equation is constructed to solve for the unknown surface area of the larger cylinder, using the following known values: - Surface area ratio: $9:25$ . - Surface area of the smaller cylinder: $378$ . - Surface area of the larger cylinder: $x$ (the variable to be solved).
The resulting proportional equation is: - $\frac{9}{25} = \frac{378}{x}$

To isolate and solve for the surface area of the larger cylinder ( $x$ ), the cross-multiplication method is employed.
The procedural steps involve: - Multiplying the surface area of the smaller cylinder by the larger part of the surface area ratio: $25 \times 378$ . - Dividing that product by the smaller part of the surface area ratio: $9$ .
The full calculation sequence is: - $x = \frac{25 \times 378}{9}$

Question: The student asks why the original ratio (the scale factor) must be squared.
Response: The instructor clarifies that because the starting value is a linear scale factor and the problem specifically asks for surface area, one must determine the ratio of the surface areas first. This conversion is achieved by squaring the linear dimensions to move from a one-dimensional scale (length) to a two-dimensional scale (area).