A cylindrical vessel 16 cm high and 9 cm as the radius of the base, is filled with sand. This vessel is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 12 cm, the radius of its base is

Options

A. 12 cm
B. 18 cm
C. 36 cm
D. 48 cm

Answer

To solve the problem step by step, we will use the formulas for the volumes of a cylinder and a cone.

### Step 1: Calculate the Volume of the Cylindrical Vessel
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius \( r = 9 \) cm
- Height \( h = 16 \) cm

Substituting the values:
\[ V = \pi \times (9)^2 \times 16 \]
\[ V = \pi \times 81 \times 16 \]
\[ V = 1296\pi \text{ cm}^3 \]

### Step 2: Set Up the Volume of the Conical Heap
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.

Given:
- Height of the conical heap \( h = 12 \) cm
- We need to find the radius \( r \).

### Step 3: Equate the Volumes
Since the volume of sand remains the same, we can equate the volume of the cylinder to the volume of the cone:
\[ 1296\pi = \frac{1}{3} \pi r^2 \times 12 \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 1296 = \frac{1}{3} r^2 \times 12 \]
Now, simplify the right side:
\[ 1296 = 4 r^2 \]

### Step 5: Solve for \( r^2 \)
To isolate \( r^2 \), multiply both sides by \( \frac{1}{4} \):
\[ r^2 = \frac{1296}{4} \]
\[ r^2 = 324 \]

### Step 6: Find \( r \)
Taking the square root of both sides gives us:
\[ r = \sqrt{324} \]
\[ r = 18 \text{ cm} \]

### Conclusion
The radius of the base of the conical heap is \( 18 \) cm.

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