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To solve this problem, we need to use the relationship between the surface areas and volumes of geometrically similar shapes. When two shapes are similar, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions (scale factor), and the ratio of their volumes is the cube of the scale factor.

### Step-by-Step Solution:

1. **Calculate the scale factor for the surface areas:**

   The total surface area of shape \(P\) is 720 cm², and the total surface area of shape \(Q\) is 2880 cm². The ratio of their surface areas is:

   \[
   \text{Surface area ratio} = \frac{\text{Area of } Q}{\text{Area of } P} = \frac{2880}{720} = 4
   \]

   Since the surface area ratio is the square of the scale factor, let \( k \) be the scale factor for their corresponding linear dimensions. Then:

   \[
   k^2 = 4
   \]

   Solving for \( k \):

   \[
   k = \sqrt{4} = 2
   \]

   So, the scale factor is 2.

2. **Use the scale factor to find the volume ratio:**

   The ratio of the volumes of two similar shapes is the cube of the scale factor. Therefore:

   \[
   \text{Volume ratio} = k^3 = 2^3 = 8
   \]

3. **Calculate the volume of shape \(Q\):**

   We know the volume of shape \(P\) is 3200 cm³. Let \( V_Q \) represent the volume of shape \(Q\). Using the volume ratio:

   \[
   \frac{V_Q}{V_P} = 8 \implies V_Q = 8 \times V_P
   \]

   Substituting the volume of shape \(P\):

   \[
   V_Q = 8 \times 3200 = 25600 \, \text{cm}^3
   \]

### Conclusion:

The volume of shape \(Q\) is **25600 cm³**.

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:

1. How do the surface area and volume ratios change if the scale factor is 3 instead of 2?
2. Can this method be used to find the dimensions of an unknown shape if the volume is known?
3. How would the volume change if shape \(Q\) had a scale factor of 1.5 compared to shape \(P\)?
4. What is the formula for the surface area and volume of a sphere? How do these formulas relate to the concept of similar shapes?
5. How would you apply this concept to find the dimensions of a smaller shape if only the dimensions of a larger shape are known?

**Tip:** Remember that for geometrically similar shapes, knowing one dimension and the scale factor allows you to determine all corresponding dimensions, areas, and volumes.

Explore how scale factors affect the surface area and volume ratios of geometrically similar shapes. Learn step-by-step how to calculate these ratios using real-world examples. Suitable for high school students and those interested in advanced geometry concepts.

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