A girl empties a cylindrical bucket full of sand, of base radius 18 cm and height 32 cm on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct to one place of decimal.

Answer

To solve the problem step by step, we will first find the volume of the cylindrical bucket and then use that volume to find the radius of the conical heap. Finally, we will calculate the slant height of the cone.

### Step 1: Calculate the volume of the cylindrical bucket
The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cylinder
- \( h \) is the height of the cylinder

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

Substituting the values:

\[ V = \pi (18)^2 (32) \]

Calculating \( (18)^2 \):

\[ (18)^2 = 324 \]

Now substituting back:

\[ V = \pi \times 324 \times 32 \]

Calculating \( 324 \times 32 \):

\[ 324 \times 32 = 10368 \]

Thus,

\[ V = 10368\pi \, \text{cm}^3 \]

### Step 2: Set the volume of the cone equal to the volume of the cylinder
The volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone
- \( h \) is the height of the cone

We know the height of the conical heap \( h = 24 \) cm. Setting the volumes equal:

\[ 10368\pi = \frac{1}{3} \pi r^2 (24) \]

### Step 3: Simplify the equation
Dividing both sides by \( \pi \):

\[ 10368 = \frac{1}{3} r^2 (24) \]

Multiplying both sides by 3:

\[ 3 \times 10368 = r^2 (24) \]

Calculating \( 3 \times 10368 \):

\[ 3 \times 10368 = 31104 \]

So we have:

\[ 31104 = r^2 (24) \]

Dividing both sides by 24:

\[ r^2 = \frac{31104}{24} \]

Calculating \( \frac{31104}{24} \):

\[ r^2 = 1296 \]

Taking the square root to find \( r \):

\[ r = \sqrt{1296} = 36 \, \text{cm} \]

### Step 4: Calculate the slant height of the cone
The slant height \( l \) of a cone can be calculated using the Pythagorean theorem:

\[ l = \sqrt{r^2 + h^2} \]

Where:
- \( r = 36 \) cm
- \( h = 24 \) cm

Substituting the values:

\[ l = \sqrt{(36)^2 + (24)^2} \]

Calculating \( (36)^2 \) and \( (24)^2 \):

\[ (36)^2 = 1296 \]
\[ (24)^2 = 576 \]

Now substituting back:

\[ l = \sqrt{1296 + 576} \]

Calculating \( 1296 + 576 \):

\[ 1296 + 576 = 1872 \]

So we have:

\[ l = \sqrt{1872} \]

Calculating \( \sqrt{1872} \):

\[ l \approx 43.2 \, \text{cm} \]

### Final Answer:
The slant height of the conical heap is approximately \( 43.2 \) cm.

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