A bucket is in the form of a frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are 14 cm and 21 cm respectively. Find the height of the bucket.

Options

A. 29.2 cm
B. 10 cm
C. 20.9 cm
D. 25 cm

Answer

To find the height of the bucket in the form of a frustum of a cone, we can follow these steps:

### Step 1: Convert the volume from liters to cubic centimeters
Since the volume of the bucket is given in liters, we need to convert it to cubic centimeters (cc) because the radii are in centimeters.

1 liter = 1000 cubic centimeters.

So,
\[ 28.490 \text{ liters} = 28.490 \times 1000 \text{ cc} = 28490 \text{ cc} \]

### Step 2: Use the formula for the volume of a frustum of a cone
The volume \( V \) of a frustum of a cone is given by the formula:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]
where:
- \( r_1 \) is the radius of the top of the frustum,
- \( r_2 \) is the radius of the bottom of the frustum,
- \( h \) is the height of the frustum.

Given:
- \( r_1 = 14 \) cm,
- \( r_2 = 21 \) cm,
- \( V = 28490 \) cc.

### Step 3: Substitute the values into the volume formula
Substituting the known values into the volume formula:
\[ 28490 = \frac{1}{3} \pi h (14^2 + 21^2 + 14 \cdot 21) \]

### Step 4: Calculate \( 14^2 \), \( 21^2 \), and \( 14 \cdot 21 \)
Calculating these values:
\[ 14^2 = 196, \]
\[ 21^2 = 441, \]
\[ 14 \cdot 21 = 294. \]

Now, substituting these into the formula:
\[ 28490 = \frac{1}{3} \pi h (196 + 441 + 294) \]
\[ 28490 = \frac{1}{3} \pi h (931). \]

### Step 5: Simplify the equation
Now, we can simplify the equation:
\[ 28490 = \frac{931\pi h}{3}. \]

### Step 6: Solve for \( h \)
To isolate \( h \), multiply both sides by 3:
\[ 3 \times 28490 = 931 \pi h, \]
\[ 85470 = 931 \pi h. \]

Now, divide both sides by \( 931 \pi \):
\[ h = \frac{85470}{931 \pi}. \]

### Step 7: Calculate the value of \( h \)
Using \( \pi \approx 3.14 \):
\[ h \approx \frac{85470}{931 \times 3.14} \approx \frac{85470}{2925.74} \approx 29.2 \text{ cm}. \]

### Final Answer
The height of the bucket is approximately \( 29.2 \) cm.

---