We have two mathematically similar solids (spheres), and we are given information about their surface areas and volumes. The task is to calculate the volume of the smaller solid.

To solve this, let's use the following relationships based on the properties of similar solids:

1. Surface Area Ratio: For two similar solids, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions (like the radius): $\frac{\text{Surface Area of smaller}}{\text{Surface Area of larger}} = \left( \frac{\text{Radius of smaller}}{\text{Radius of larger}} \right)^2$

2. Volume Ratio: The ratio of their volumes is the cube of the ratio of their corresponding linear dimensions: $\frac{\text{Volume of smaller}}{\text{Volume of larger}} = \left( \frac{\text{Radius of smaller}}{\text{Radius of larger}} \right)^3$

Step 1: Find the ratio of the radii

The ratio of the surface areas is: $\frac{420}{1500} = \left( \frac{r_s}{r_l} \right)^2$ where $r_s$ and $r_l$ are the radii of the smaller and larger solid, respectively. Solving for $\frac{r_s}{r_l}$, we get: $\frac{r_s}{r_l} = \sqrt{\frac{420}{1500}} = \sqrt{0.28}$ $\frac{r_s}{r_l} \approx 0.529$

Step 2: Find the ratio of the volumes

Next, the ratio of the volumes is: $\frac{V_s}{V_l} = \left( \frac{r_s}{r_l} \right)^3$ Substitute the value of $\frac{r_s}{r_l}$: $\frac{V_s}{V_l} = (0.529)^3 \approx 0.148$

Step 3: Calculate the volume of the smaller solid

Now, we can find the volume of the smaller solid by using the known volume of the larger solid: $\frac{V_s}{67} = 0.148$ $V_s = 67 \times 0.148 \approx 9.92 \, \text{cm}^3$

Final Answer:

The volume of the smaller solid is approximately $\boxed{9.92} \, \text{cm}^3$ to three significant figures.

Would you like further details or clarifications? Here are some related questions:

1. How can you apply the same approach to find the ratio of two similar pyramids' volumes?
2. What would happen to the volume of a solid if the surface area were doubled while maintaining similarity?
3. How do we calculate the surface area of a sphere given its radius?
4. How does the ratio of surface areas relate to the ratio of volumes for similar solids?
5. How could this problem be extended to other 3D shapes like cones or cylinders?

Tip: For geometrically similar solids, the ratio of surface areas and volumes always involves powers of the linear dimension ratio.