The curved surface area of a cone is 550cm sq . If the area of its base is 154 cm sq , then what will be the volume of the cone?

Options

A. `2464cm^(3)`
B. `1836cm^(3)`
C. `1232cm^(3)`
D. `1472cm^(3)`

Answer

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

### Step 1: Understand the given information
We know:
- Curved Surface Area (CSA) of the cone = 550 cm²
- Area of the base = 154 cm²

### Step 2: Use the formula for the curved surface area
The formula for the curved surface area of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height of the cone.

### Step 3: Substitute the values into the CSA formula
We can substitute the CSA value into the formula:
\[ \pi r l = 550 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r l = 550 \]

### Step 4: Calculate the area of the base to find the radius
The area of the base of the cone is given by:
\[ \text{Area of base} = \pi r^2 \]
Substituting the area of the base:
\[ \pi r^2 = 154 \]
Again using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 154 \]

### Step 5: Solve for \( r^2 \)
To isolate \( r^2 \):
\[ r^2 = \frac{154 \times 7}{22} \]
Calculating:
\[ r^2 = \frac{1078}{22} = 49 \]
Thus, \( r = \sqrt{49} = 7 \) cm.

### Step 6: Substitute \( r \) back to find \( l \)
Now we substitute \( r \) back into the CSA formula to find \( l \):
\[ \frac{22}{7} \times 7 \times l = 550 \]
This simplifies to:
\[ 22l = 550 \]
So,
\[ l = \frac{550}{22} = 25 \text{ cm} \]

### Step 7: Use the Pythagorean theorem to find the height \( h \)
We know that:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + h^2 \]
Calculating:
\[ 625 = 49 + h^2 \]
Thus,
\[ h^2 = 625 - 49 = 576 \]
So,
\[ h = \sqrt{576} = 24 \text{ cm} \]

### Step 8: Calculate the volume of the cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 7^2 \times 24 \]
This simplifies to:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 24 \]
Calculating:
\[ V = \frac{1}{3} \times 22 \times 7 \times 24 \]
\[ V = \frac{1}{3} \times 3696 = 1232 \text{ cm}^3 \]

### Final Answer
The volume of the cone is \( 1232 \text{ cm}^3 \).

---