We are given two similar cylinders, and we need to find the surface area of the smaller cylinder. The provided information includes:

• The height of the smaller cylinder = 3 cm.
• The height of the larger cylinder = 5 cm.
• The surface area of the larger cylinder = 236 cm².

Since the cylinders are similar, their surface areas are proportional to the square of the ratio of their heights. The ratio of their heights is:

$\text{Height ratio} = \frac{3 \, \text{cm}}{5 \, \text{cm}} = \frac{3}{5}.$

Now, the ratio of the surface areas will be the square of this ratio:

$\text{Surface area ratio} = \left(\frac{3}{5}\right)^2 = \frac{9}{25}.$

Using this ratio, we can now find the surface area of the smaller cylinder by multiplying the surface area of the larger cylinder by the ratio:

$\text{Surface area of smaller cylinder} = 236 \, \text{cm}^2 \times \frac{9}{25}.$

Let me calculate that.The surface area of the smaller cylinder is approximately $85.0 \, \text{cm}^2$ when rounded to the nearest tenth.

Do you want further details or have any questions?

Here are 5 related questions for further exploration:

1. How do we derive the surface area formula for a cylinder?
2. If the heights were different, how would the solution change?
3. How can we apply the concept of similarity to other 3D shapes?
4. What happens to the surface area if both the height and radius change proportionally?
5. How do we find the volume of similar cylinders based on their height ratios?

Tip: When working with similar shapes, remember that surface area scales with the square of the ratio of corresponding lengths, while volume scales with the cube.