Total volume of three identical cones is the same as that of a bigger cone whose height is 9 cm and diameter 40 cm. Find the radius of the base of each smaller cone, if height of each is 108 cm.

Answer

To solve the problem, we need to find the radius of the base of each smaller cone given that the total volume of three identical smaller cones is equal to the volume of a larger cone.

### Step-by-Step Solution:

1. **Identify the dimensions of the larger cone:**
- Height (H) = 9 cm
- Diameter = 40 cm
- Radius (R) = Diameter / 2 = 40 cm / 2 = 20 cm

2. **Calculate the volume of the larger cone using the formula for the volume of a cone:**
\[ V = \frac{1}{3} \pi R^2 H \]
Substituting the values:
\[ V = \frac{1}{3} \pi (20)^2 (9) = \frac{1}{3} \pi (400) (9) = \frac{3600}{3} \pi = 1200 \pi \text{ cm}^3 \]

3. **Identify the dimensions of the smaller cones:**
- Number of smaller cones = 3
- Height (h) = 108 cm
- Let the radius of each smaller cone be \( r \).

4. **Set up the equation for the total volume of the three smaller cones:**
The volume of one smaller cone is:
\[ V_{small} = \frac{1}{3} \pi r^2 h \]
Therefore, the total volume of three smaller cones is:
\[ 3 \times V_{small} = 3 \times \frac{1}{3} \pi r^2 (108) = \pi r^2 (108) \]

5. **Set the total volume of the smaller cones equal to the volume of the larger cone:**
\[ \pi r^2 (108) = 1200 \pi \]

6. **Cancel \(\pi\) from both sides:**
\[ r^2 (108) = 1200 \]

7. **Solve for \(r^2\):**
\[ r^2 = \frac{1200}{108} \]

8. **Simplify \( \frac{1200}{108} \):**
\[ r^2 = \frac{1200 \div 12}{108 \div 12} = \frac{100}{9} \]

9. **Take the square root to find \(r\):**
\[ r = \sqrt{\frac{100}{9}} = \frac{10}{3} \text{ cm} \]

### Final Answer:
The radius of the base of each smaller cone is \( \frac{10}{3} \) cm.