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Given that vases \( A \) and \( B \) are similar, we know that their volumes are proportional to the cube of their corresponding linear dimensions. The height of vase \( A \) is \( 10 \, \text{cm} \) and the height of vase \( B \) is \( 15 \, \text{cm} \). 

### Step 1: Find the scale factor
The scale factor \( k \) from vase \( A \) to vase \( B \) is:
\[
k = \frac{\text{Height of B}}{\text{Height of A}} = \frac{15}{10} = 1.5
\]

### Step 2: Apply the scale factor to find the volume ratio
Since the vases are similar, the ratio of their volumes is the cube of the linear scale factor:
\[
\left(\frac{\text{Volume of B}}{\text{Volume of A}}\right) = k^3 = (1.5)^3 = 3.375
\]

Let \( V_A \) represent the volume of vase \( A \) and \( V_B \) represent the volume of vase \( B \). Then:
\[
V_B = 3.375 \times V_A
\]

### Step 3: Set up and solve the equation
We are given that the difference in volumes is \( 1197 \, \text{cm}^3 \), so:
\[
V_B - V_A = 1197
\]

Substitute \( V_B = 3.375 \times V_A \):
\[
3.375 \, V_A - V_A = 1197
\]
\[
(3.375 - 1) \, V_A = 1197
\]
\[
2.375 \, V_A = 1197
\]
\[
V_A = \frac{1197}{2.375} = 504 \, \text{cm}^3
\]

### Answer
The volume of vase \( A \) is:
\[
\boxed{504 \, \text{cm}^3}
\]

Would you like a detailed explanation on any part of this solution? 

Here are 5 related questions:
1. What would be the volume of vase \( B \) based on our findings?
2. How does changing the height of a similar object affect its volume?
3. If the height of vase \( B \) were doubled, what would be the new volume of vase \( A \)?
4. How is the volume ratio derived from the linear scale factor?
5. Can this method be applied to other three-dimensional shapes with similar properties?

**Tip:** For similar objects, remember that the volume scales with the cube of the linear dimension ratio.

A and B are two similar vases, not drawn to scale. The difference between the volume of vase A and the volume of vase B is 1197 cm³. Calculate the volume of vase A.

Learn how to calculate the volume of a smaller vase (vase A) given the volume difference with a larger, similar vase (vase B) and their scale factor. Explore the relationship between similarity, scale factor, and volume in three-dimensional shapes. Ideal for Grades 9-11.

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