To solve this problem, we will use the fact that for two similar solids, the ratio of their volumes is the cube of the ratio of their corresponding lengths (such as the radius for cylinders).

Let $r_1$ be the radius of the smaller cylinder and $r_2 = 8 \, \text{cm}$ be the radius of the larger cylinder.

Given:

• The volume of the smaller cylinder: $V_1 = 125 \, \text{cm}^3$
• The volume of the larger cylinder: $V_2 = 512 \, \text{cm}^3$
• The ratio of the volumes is: $\frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3$

Substitute the values: $\frac{125}{512} = \left( \frac{r_1}{8} \right)^3$

Now solve for $\frac{r_1}{8}$:

$\left( \frac{r_1}{8} \right)^3 = \frac{125}{512}$ $\frac{r_1}{8} = \sqrt[3]{\frac{125}{512}}$ $\frac{r_1}{8} = \frac{5}{8}$

Finally, multiply both sides by 8 to find $r_1$:

$r_1 = 8 \times \frac{5}{8} = 5 \, \text{cm}$

Thus, the radius of the smaller cylinder is 5 cm.

Would you like further details or have any questions?

Here are some related questions:

1. How do surface areas of similar cylinders relate?
2. What if the height of the larger cylinder is given—how does that change the approach?
3. How would you calculate the height of the smaller cylinder if the heights were also proportional?
4. Can you derive the relationship between the surface area and volume of a cylinder?
5. How can you use this method for pyramids or cones?

Tip: When working with similar figures, always remember that areas scale with the square of the corresponding length ratio, while volumes scale with the cube.