The height of a cone is 21 cm and its slant height is 28 cm. The volume of the cone is

Options

A. `7356 cm^(3)`
B. `7546 cm^(3)`
C. `7506 cm^(2)`
D. `7564 cm^(3)`

Answer

To find the volume of the cone, we will follow these steps:

### Step 1: Identify the given values
- Height (h) of the cone = 21 cm
- Slant height (l) of the cone = 28 cm

### Step 2: Use the Pythagorean theorem to find the radius (r)
The relationship between the radius (r), height (h), and slant height (l) of a cone is given by the formula:
\[ l^2 = r^2 + h^2 \]

Substituting the known values:
\[ 28^2 = r^2 + 21^2 \]

Calculating the squares:
\[ 784 = r^2 + 441 \]

### Step 3: Solve for the radius (r)
Rearranging the equation to find \( r^2 \):
\[ r^2 = 784 - 441 \]
\[ r^2 = 343 \]

Taking the square root to find r:
\[ r = \sqrt{343} \]
\[ r = 18.52 \, \text{cm} \] (approximately)

### Step 4: Use the formula for the volume of the cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]

Substituting the values of \( r \) and \( h \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 343 \times 21 \]

### Step 5: Calculate the volume
First, calculate \( \frac{1}{3} \times 21 = 7 \):
\[ V = 7 \times \frac{22}{7} \times 343 \]

Now, the \( 7 \) cancels out:
\[ V = 22 \times 343 \]

Calculating:
\[ V = 7546 \, \text{cm}^3 \]

### Final Answer
The volume of the cone is \( 7546 \, \text{cm}^3 \).

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