A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, find :
(i) the radius and
(ii) the slant height of the heap. Give your answer correct to one place of decimal.

Answer

To solve the problem step by step, we will follow these calculations:

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

Given:
- Radius \( r = 18 \) cm
- Height \( h = 32 \) cm

Substituting the values:
\[ V = \pi \times (18)^2 \times 32 \]
Calculating \( (18)^2 \):
\[ (18)^2 = 324 \]
Now substituting back:
\[ V = \pi \times 324 \times 32 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 324 \times 32 \]
Calculating \( 324 \times 32 \):
\[ 324 \times 32 = 10368 \]
Now substituting:
\[ V = \frac{22}{7} \times 10368 \]
Calculating \( \frac{22 \times 10368}{7} \):
\[ 22 \times 10368 = 228096 \]
Now dividing by 7:
\[ V = \frac{228096}{7} \approx 32585.1 \text{ cm}^3 \]

### Step 2: Set the Volume of the Cone Equal to the Volume of the Cylinder
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 cone
- \( h \) is the height of the cone

Given:
- Height of the cone \( h = 24 \) cm

Setting the volumes equal:
\[ 32585.1 = \frac{1}{3} \pi r^2 \times 24 \]
Substituting \( \pi \approx \frac{22}{7} \):
\[ 32585.1 = \frac{1}{3} \times \frac{22}{7} \times r^2 \times 24 \]
Calculating \( \frac{1}{3} \times 24 = 8 \):
\[ 32585.1 = \frac{22}{7} \times 8 \times r^2 \]
Calculating \( \frac{22 \times 8}{7} = \frac{176}{7} \):
\[ 32585.1 = \frac{176}{7} r^2 \]
Multiplying both sides by 7:
\[ 228095.7 = 176 r^2 \]
Now dividing by 176:
\[ r^2 = \frac{228095.7}{176} \approx 1296 \]

### Step 3: Calculate the Radius of the Cone
Taking the square root:
\[ r = \sqrt{1296} = 36 \text{ cm} \]

### Step 4: Calculate the Slant Height of the Cone
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Where:
- \( h = 24 \) cm
- \( r = 36 \) cm

Calculating:
\[ l = \sqrt{(24)^2 + (36)^2} \]
Calculating \( (24)^2 \) and \( (36)^2 \):
\[ (24)^2 = 576, \quad (36)^2 = 1296 \]
Adding these:
\[ l = \sqrt{576 + 1296} = \sqrt{1872} \]
Calculating \( \sqrt{1872} \):
\[ l \approx 43.3 \text{ cm} \]

### Final Answers
(i) Radius of the conical heap: \( 36 \) cm
(ii) Slant height of the conical heap: \( 43.3 \) cm