To solve this problem, let’s go through it step-by-step.

1. Given Information:

   • The vases $A$ and $B$ are similar, meaning their shapes are the same, but their sizes are different.
   • The height of vase $A$ is $10 \, \text{cm}$, and the height of vase $B$ is $15 \, \text{cm}$.
   • The difference in volume between vase $B$ and vase $A$ is $1197 \, \text{cm}^3$.

2. Volume of Similar Shapes:

   • For similar shapes, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions (such as height).
   • The height ratio of vases $A$ and $B$ is: $\frac{\text{Height of } B}{\text{Height of } A} = \frac{15}{10} = \frac{3}{2}$
   • Therefore, the volume ratio will be: $\left(\frac{3}{2}\right)^3 = \frac{27}{8}$

3. Setting Up Equations:

   • Let the volume of vase $A$ be $V_A$.
   • Then the volume of vase $B$ will be: $V_B = \frac{27}{8} V_A$
   • According to the problem, the difference in volume between $B$ and $A$ is $1197 \, \text{cm}^3$: $V_B - V_A = 1197$
   • Substituting $V_B$ from above: $\frac{27}{8} V_A - V_A = 1197$

4. Solving for $V_A$:

   • Simplify the equation: $\frac{27V_A - 8V_A}{8} = 1197$ $\frac{19V_A}{8} = 1197$
   • Multiply both sides by $8$ to isolate $V_A$: $19V_A = 1197 \times 8$ $19V_A = 9576$
   • Divide by $19$ to solve for $V_A$: $V_A = \frac{9576}{19} = 504 \, \text{cm}^3$

5. Conclusion:

   • The volume of vase $A$ is $504 \, \text{cm}^3$.

Would you like further details on any specific step? Here are some related questions you might find interesting:

1. How would the volume change if the heights of the vases were doubled?
2. What is the volume of vase $B$?
3. Can this method be applied to other similar 3D objects, like cones or pyramids?
4. How would the calculations change if only the width of the vases differed?
5. What is the significance of the volume ratio in real-life applications?

Tip: When dealing with similar shapes, remember that areas scale by the square of the ratio, while volumes scale by the cube of the ratio.