Question

Do you think doubling the radius of a cylinder (but keeping the same height) will change the surface area more or change the volume more? Show an example to support your thinking.

Do you think doubling the radius of a cylinder or doubling the height will affect the surface area more? How about the volume? Show an example to support your thinking.

Answer 1

Doubling the radius of a cylinder (while keeping height constant) increases the surface area more significantly than the volume. Doubling the height affects surface area more, while doubling the radius affects volume more.

Doubling the radius of a cylinder while keeping the height constant will affect the surface area more than the volume. The surface area A and volume V formulas for a cylinder are:

`\[ A = 2\pi r^2 + 2\pi rh \]\[ V = \pi r^2h \]`

If we double the radius
`(\(r \rightarrow 2r\)),` the new surface area becomes:

`\[ A' = 2\pi (2r)^2 + 2\pi (2r)h = 8\pi r^2 + 4\pi rh \]`

And the new volume becomes:

`\[ V' = \pi (2r)^2h = 4\pi r^2h \]`

Comparing the changes, we see that the surface area increases by a factor of
`\(8\pi\)` (more significant) while the volume increases by a factor of
`\(4\pi\)` (less significant).

Now, for doubling the radius versus doubling the height, consider doubling the radius
`(\(r \rightarrow 2r\))`:

`\[ A' = 2\pi (2r)^2 + 2\pi (2r)h = 8\pi r^2 + 4\pi rh \]\[ V' = \pi (2r)^2h = 4\pi r^2h \]`

And doubling the height
`(\(h \rightarrow 2h\))`:

`\[ A'' = 2\pi r^2 + 2\pi r(2h) = 2\pi r^2 + 4\pi rh \]\[ V'' = \pi r^2(2h) = 2\pi r^2h \]`

Comparing the changes, doubling the height has a greater impact on the surface area, increasing it by a factor of
`\(2\pi\)` (more significant), while doubling the radius has a greater impact on the volume, increasing it by a factor of
`\(4\pi\)` (more significant).