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.