A hemisphere of lead of radius 9 cm is cast into a right circular cone of height 72 cm. Find the radius of the base of the cone.

Explanation

To solve the problem, we need to find the radius of the base of the cone formed by casting a hemisphere of lead. We will use the volume formulas for both the hemisphere and the cone.

### Step-by-Step Solution:

1. **Calculate the Volume of the Hemisphere:**
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere. Here, \( r = 9 \) cm.
\[ V = \frac{2}{3} \pi (9)^3 \]
\[ V = \frac{2}{3} \pi (729) \]
\[ V = \frac{1458}{3} \pi = 486 \pi \text{ cm}^3 \]

2. **Set Up the Volume of the Cone:**
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi R^2 h \]
where \( R \) is the radius of the base of the cone and \( h \) is the height of the cone. Here, \( h = 72 \) cm and we need to find \( R \).

3. **Equate the Volumes:**
Since the volume of the lead remains the same when it is cast into the cone, we can set the volume of the hemisphere equal to the volume of the cone:
\[ 486 \pi = \frac{1}{3} \pi R^2 (72) \]

4. **Simplify the Equation:**
We can cancel \( \pi \) from both sides:
\[ 486 = \frac{1}{3} R^2 (72) \]
Multiply both sides by 3 to eliminate the fraction:
\[ 1458 = R^2 (72) \]

5. **Solve for \( R^2 \):**
Divide both sides by 72:
\[ R^2 = \frac{1458}{72} \]
Simplifying \( \frac{1458}{72} \):
\[ R^2 = 20.25 \]

6. **Find \( R \):**
Take the square root of both sides to find \( R \):
\[ R = \sqrt{20.25} = 4.5 \text{ cm} \]

### Final Answer:
The radius of the base of the cone is \( 4.5 \) cm.