We are given two shapes, A and B, that are mathematically similar. The surface areas of their bases are provided, and we are tasked with determining the volume of shape B. Shape A’s volume is already given as $\frac{250\pi}{3} \, \text{cm}^3$.

Approach:

1. Surface Area Relationship: Since the two shapes are similar, the ratio of their corresponding areas is proportional to the square of the ratio of their corresponding linear dimensions (scale factor). The surface areas of their bases are:

   • Shape A: $25\pi \, \text{cm}^2$
   • Shape B: $49\pi \, \text{cm}^2$

   The ratio of their base surface areas is: $\frac{S_B}{S_A} = \frac{49\pi}{25\pi} = \frac{49}{25}$

   This is the square of the ratio of their linear dimensions (scale factor), so the scale factor $k$ is: $k = \sqrt{\frac{49}{25}} = \frac{7}{5}$

2. Volume Relationship: The volumes of similar shapes scale according to the cube of the linear scale factor. Therefore, the ratio of the volumes of shapes B and A is: $\frac{V_B}{V_A} = \left( \frac{7}{5} \right)^3 = \frac{343}{125}$

   Now, using the volume of shape A, $V_A = \frac{250\pi}{3} \, \text{cm}^3$, we can calculate the volume of shape B as follows: $V_B = V_A \times \frac{343}{125} = \frac{250\pi}{3} \times \frac{343}{125}$

   Simplifying: $V_B = \frac{250 \times 343 \pi}{3 \times 125} = \frac{85750 \pi}{375} = \frac{343 \pi}{3} \, \text{cm}^3$

Thus, the volume of shape B is $\frac{343 \pi}{3} \, \text{cm}^3$.

Would you like a further breakdown of any step or have additional questions?

Here are 5 related questions:

1. What is the relationship between surface area and volume scaling in similar shapes?
2. How would the formula change if we were given the heights of the shapes instead of the areas?
3. Can this method be used for other types of shapes like pyramids or cones?
4. How can we find the height of shape B using the given surface area and volume?
5. What if the surface area of one shape were unknown—how could we approach this?

Tip: Always remember that when dealing with similar shapes, the ratio of areas scales with the square of the linear dimensions, and the ratio of volumes scales with the cube!