When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$.  When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$.  What is the length (in cm) of the hypotenuse of the triangle?
1. Let the legs of the right triangle be \( x \) and \( y \). We need to find the length of the hypotenuse, which is \( \sqrt{x^2 + y^2} \).

2. When the triangle is rotated about one leg (say \( x \)), the volume of the cone produced is given by:
   \[
   V_1 = \frac{1}{3} \pi x y^2 = 800 \pi \text{ cm}^3
   \]
   Simplifying, we get:
   \[
   \frac{1}{3} x y^2 = 800 \implies x y^2 = 2400
   \]

3. When the triangle is rotated about the other leg (say \( y \)), the volume of the cone produced is given by:
   \[
   V_2 = \frac{1}{3} \pi y x^2 = 1920 \pi \text{ cm}^3
   \]
   Simplifying, we get:
   \[
   \frac{1}{3} y x^2 = 1920 \implies y x^2 = 5760
   \]

4. We now have the system of equations:
   \[
   x y^2 = 2400 \quad \text{(1)}
   \]
   \[
   y x^2 = 5760 \quad \text{(2)}
   \]

5. To find \( x \) and \( y \), we multiply the two equations together:
   \[
   (x y^2) (y x^2) = 2400 \cdot 5760
   \]
   Simplifying, we get:
   \[
   x^3 y^3 = 2400 \cdot 5760
   \]
   Taking the cube root of both sides:
   \[
   (xy)^3 = 2400 \cdot 5760
   \]
   \[
   xy = \sqrt[3]{2400 \cdot 5760}
   \]

6. Calculate \( 2400 \cdot 5760 \):
   \[
   2400 \cdot 5760 = 13824000
   \]
   Taking the cube root:
   \[
   xy = \sqrt[3]{13824000} = 240
   \]

7. Now we have \( xy = 240 \). Substitute \( y = \frac{240}{x} \) into equation (1):
   \[
   x \left( \frac{240}{x} \right)^2 = 2400
   \]
   Simplifying:
   \[
   x \cdot \frac{57600}{x^2} = 2400
   \]
   \[
   \frac{57600}{x} = 2400
   \]
   Solving for \( x \):
   \[
   x = \frac{57600}{2400} = 24
   \]

8. Substitute \( x = 24 \) back into \( y = \frac{240}{x} \):
   \[
   y = \frac{240}{24} = 10
   \]

9. Finally, calculate the hypotenuse:
   \[
   \sqrt{x^2 + y^2} = \sqrt{24^2 + 10^2} = \sqrt{576 + 100} = \sqrt{676} = 26
   \]

The final answer is \( \boxed{26} \) cm.