If the radius of the base of a cone be 7 cm and its curved surface area be 550 sq. cm, then the volume of the cone is

Options

A. 1232 cu. cm
B. 1024 cu. cm
C. 1132 cu. cm
D. 1324 cu. cm

Answer

To find the volume of the cone given the radius and the curved surface area, we can follow these steps:

### Step 1: Understand the formula for the curved surface area of a cone.
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius of the base and \( l \) is the slant height.

### Step 2: Substitute the known values into the formula.
Given:
- Radius \( r = 7 \) cm
- Curved Surface Area \( \text{CSA} = 550 \) sq. cm

We can substitute these values into the formula:
\[ 550 = \pi \times 7 \times l \]

### Step 3: Solve for the slant height \( l \).
Using \( \pi \approx \frac{22}{7} \):
\[ 550 = \frac{22}{7} \times 7 \times l \]
This simplifies to:
\[ 550 = 22l \]

Now, divide both sides by 22:
\[ l = \frac{550}{22} = 25 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find the height \( h \).
The relationship between the height \( h \), radius \( r \), and slant height \( l \) is given by:
\[ l^2 = h^2 + r^2 \]

Substituting the known values:
\[ 25^2 = h^2 + 7^2 \]
\[ 625 = h^2 + 49 \]

### Step 5: Solve for \( h^2 \).
Rearranging gives:
\[ h^2 = 625 - 49 \]
\[ h^2 = 576 \]

### Step 6: Find the height \( h \).
Taking the square root:
\[ h = \sqrt{576} = 24 \text{ cm} \]

### Step 7: Calculate 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 known values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 7^2 \times 24 \]
\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 24 \]

### Step 8: Simplify the volume calculation.
Calculating step by step:
1. Calculate \( 49 \times 24 = 1176 \)
2. Now substitute back:
\[ V = \frac{1}{3} \times \frac{22 \times 1176}{7} \]
3. Calculate \( 22 \times 1176 = 25872 \)
4. Now divide by 7:
\[ \frac{25872}{7} = 3696 \]
5. Finally, divide by 3:
\[ V = \frac{3696}{3} = 1232 \text{ cubic cm} \]

### Final Answer:
The volume of the cone is \( 1232 \) cubic cm.

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