A cylindrical container of 32 cm height and 18 cm radius is filled with sand. Now all this sand is used to form a conical heap of sand. If the height of the conical heap is 24 cm, what is he radius of its base ?

Options

A. a.12cm
B. b.24 cm
C. c.36 cm
D. d.48 cm

Answer

To solve the problem, we need to find the radius of the base of a conical heap of sand formed from the sand in a cylindrical container. We will use the formula for the volumes of a cylinder and a cone.

### Step-by-Step Solution:

1. **Identify the dimensions of the cylindrical container**:
- Height (h_cylinder) = 32 cm
- Radius (r_cylinder) = 18 cm

2. **Calculate the volume of the cylindrical container**:
The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi r^2 h \]
Substituting the values:
\[ V_{cylinder} = \pi (18)^2 (32) \]
\[ V_{cylinder} = \pi (324) (32) \]
\[ V_{cylinder} = 10368\pi \, \text{cm}^3 \]

3. **Identify the dimensions of the conical heap**:
- Height (h_cone) = 24 cm
- Radius (r_cone) = ? (This is what we need to find)

4. **Set up the equation for the volume of the cone**:
The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Since the volume of the sand remains the same, we can set the volumes equal:
\[ V_{cylinder} = V_{cone} \]
\[ 10368\pi = \frac{1}{3} \pi r_{cone}^2 (24) \]

5. **Simplify the equation**:
Cancel \(\pi\) from both sides:
\[ 10368 = \frac{1}{3} r_{cone}^2 (24) \]
Multiply both sides by 3 to eliminate the fraction:
\[ 31024 = r_{cone}^2 (24) \]
Divide both sides by 24:
\[ r_{cone}^2 = \frac{31024}{24} \]
\[ r_{cone}^2 = 1296 \]

6. **Calculate the radius of the conical heap**:
To find \(r_{cone}\), take the square root of both sides:
\[ r_{cone} = \sqrt{1296} \]
\[ r_{cone} = 36 \, \text{cm} \]

### Final Answer:
The radius of the base of the conical heap is **36 cm**.

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