Question

A cylinder with a radius of 4cm and a height of 8cm has a surface area of 96pi cm^2 if another similar cylinder has a scale factor of 1/2 to this cylinder what is the surface area in terms of pi for that cylinder in cm^2

Answer 1

The surface area \( A \) of a cylinder is calculated using the formula:

\[
A = 2\pi r^2 + 2\pi rh
\]

where \( r \) is the radius and \( h \) is the height of the cylinder.

For the original cylinder, we already know the surface area is \( 96\pi \) cm², with a radius \( r = 4 \) cm and height \( h = 8 \) cm.

Now, if another similar cylinder has a scale factor of \( \frac{1}{2} \), the dimensions of the new cylinder will be:

- New radius \( r' = \frac{1}{2} \times 4 \, \text{cm} = 2 \, \text{cm} \)
- New height \( h' = \frac{1}{2} \times 8 \, \text{cm} = 4 \, \text{cm} \)

Next, we calculate the surface area of the new cylinder. Plugging the new dimensions into the surface area formula:

\[
A' = 2\pi (r')^2 + 2\pi (r')(h')
\]

Substituting \( r' = 2 \) cm and \( h' = 4 \) cm:

\[
A' = 2\pi (2)^2 + 2\pi (2)(4)
\]
\[
= 2\pi (4) + 2\pi (8)
\]
\[
= 8\pi + 16\pi
\]
\[
= 24\pi \, \text{cm}^2
\]

Thus, the surface area of the smaller cylinder in terms of \( \pi \) is

\[
\boxed{24\pi} \, \text{cm}^2.
\]