To solve this problem, we need to use the concept of scale factors between similar shapes. Since Shapes A and B are mathematically similar, the ratio of their areas and volumes can be related by certain scale factors.

Given:

• Volume of Shape A ($V_A$) = 14 cm³
• Base surface area of Shape A ($S_A$) = 10 cm²
• Base surface area of Shape B ($S_B$) = 57.6 cm²

Step-by-step Solution:

1. Find the Area Scale Factor:
   The ratio of the surface areas gives the square of the linear scale factor.
   $\text{Area scale factor} = \frac{S_B}{S_A} = \frac{57.6}{10} = 5.76$

2. Find the Linear Scale Factor:
   Since the area scale factor is the square of the linear scale factor, we can find the linear scale factor by taking the square root: $\text{Linear scale factor} = \sqrt{5.76} = 2.4$

3. Find the Volume Scale Factor:
   The volume scale factor is the cube of the linear scale factor: $\text{Volume scale factor} = (2.4)^3 = 13.824$

4. Calculate the Volume of Shape B:
   The volume of Shape B can be found by multiplying the volume of Shape A by the volume scale factor: $V_B = V_A \times 13.824 = 14 \times 13.824 = 193.536 \, \text{cm}^3$

Thus, the volume of Shape B is 193.536 cm³.

Would you like more details or have any questions?

Related Questions:

1. What is the difference between surface area and volume scale factors for similar shapes?
2. How do you derive the relationship between linear, area, and volume scale factors?
3. Why do we cube the linear scale factor to find the volume scale factor?
4. How do you apply this concept to non-cylindrical shapes?
5. Can this method be used to find the surface area if the volumes are known?

Tip:

When working with similar shapes, remember that areas scale by the square of the linear scale factor, while volumes scale by the cube of the linear scale factor.