A cylindrical bucket of height 36 cm and radius 21 cm is filled with sand. The bucket is emptied on the ground and a conical heap of sand is formed, the height of the heap being 12 cm. The radius of the heap at the base is:

Options

A. 63cm
B. 53cm
C. 56cm
D. 66cm

Answer

To find the radius of the conical heap of sand formed when the cylindrical bucket is emptied, we can follow these steps:

### Step 1: Calculate the Volume of the Cylinder
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:
- Height of the cylinder \( h = 36 \) cm
- Radius of the cylinder \( r = 21 \) cm

Substituting the values into the formula:
\[ V = \pi (21)^2 (36) \]
Calculating \( (21)^2 \):
\[ (21)^2 = 441 \]
Now substituting back:
\[ V = \pi \times 441 \times 36 \]
Calculating \( 441 \times 36 \):
\[ 441 \times 36 = 15996 \]
Thus, the volume of the cylinder is:
\[ V = 15996\pi \text{ cm}^3 \]

### Step 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.

Given:
- Height of the cone \( H = 12 \) cm

Since the volume of the sand remains the same, we can equate the volume of the cylinder to the volume of the cone:
\[ 15996\pi = \frac{1}{3} \pi R^2 (12) \]

### Step 3: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 15996 = \frac{1}{3} R^2 (12) \]
This simplifies to:
\[ 15996 = 4 R^2 \]

### Step 4: Solve for \( R^2 \)
To isolate \( R^2 \), we multiply both sides by \( \frac{1}{4} \):
\[ R^2 = \frac{15996}{4} \]
Calculating \( \frac{15996}{4} \):
\[ R^2 = 3999 \]

### Step 5: Calculate \( R \)
Now, we take the square root of both sides to find \( R \):
\[ R = \sqrt{3999} \]
Calculating \( \sqrt{3999} \) gives approximately:
\[ R \approx 63.245 \]
Rounding to the nearest whole number, we find:
\[ R \approx 63 \text{ cm} \]

### Conclusion
The radius of the conical heap of sand is approximately \( 63 \) cm.

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