Question

A cylinder with a radius of 4 cm and height of 8 cm has a surface area of 96π cm2 if another similar cylinder has a scale factor of 1/2 to this cylinder what is the surface area in terms of π for that cylinder in cm2 _____πcm2

Answer 1

To find the surface area of the smaller cylinder with a scale factor of \( \frac{1}{2} \), we start with the surface area of the original cylinder.

The surface area (SA) of a cylinder is given by the formula:
\[
SA = 2\pi r^2 + 2\pi rh
\]
where \( r \) is the radius and \( h \) is the height.

For the original cylinder:
- Radius \( r = 4 \) cm
- Height \( h = 8 \) cm

Calculating the surface area:
\[
SA = 2\pi (4)^2 + 2\pi (4)(8)
\]
\[
SA = 2\pi (16) + 2\pi (32)
\]
\[
SA = 32\pi + 64\pi = 96\pi \, \text{cm}^2
\]

Now, let’s consider the smaller cylinder, which has a scale factor of \( \frac{1}{2} \). This means that the radius and height of the smaller cylinder will be:
\[
\text{New radius} = \frac{1}{2} \times 4 = 2 \, \text{cm}
\]
\[
\text{New height} = \frac{1}{2} \times 8 = 4 \, \text{cm}
\]

Next, we calculate the surface area of the smaller cylinder using the same formula:
\[
SA_{\text{smaller}} = 2\pi (2)^2 + 2\pi (2)(4)
\]
\[
SA_{\text{smaller}} = 2\pi (4) + 2\pi (8)
\]
\[
SA_{\text{smaller}} = 8\pi + 16\pi = 24\pi \, \text{cm}^2
\]

Thus, the surface area of the smaller cylinder in terms of π is:
\[
\boxed{24\pi} \, \text{cm}^2
\]