The height of a cone is 24 cm and the area of the bases is `154 cm^(2)`. What is the curved surface area `("in" cm^(2))` of the cone ?

Options

A. 484
B. 550
C. 525
D. 515

Answer

To find the curved surface area (CSA) of the cone, we will follow these steps:

### Step 1: Understand the given information
- Height of the cone (h) = 24 cm
- Area of the base (A) = 154 cm²

### Step 2: Relate the area of the base to the radius
The area of the base of a cone is given by the formula:
\[ A = \pi r^2 \]
Where \( r \) is the radius of the base.

### Step 3: Solve for the radius
We know that:
\[ 154 = \pi r^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ 154 = \frac{22}{7} r^2 \]
To eliminate the fraction, multiply both sides by 7:
\[ 154 \times 7 = 22 r^2 \]
\[ 1078 = 22 r^2 \]
Now divide both sides by 22:
\[ r^2 = \frac{1078}{22} \]
Calculating this gives:
\[ r^2 = 49 \]
Taking the square root of both sides, we find:
\[ r = 7 \text{ cm} \]

### Step 4: Calculate the slant height (l)
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the known values:
\[ l = \sqrt{24^2 + 7^2} \]
Calculating the squares:
\[ l = \sqrt{576 + 49} \]
\[ l = \sqrt{625} \]
Thus,
\[ l = 25 \text{ cm} \]

### Step 5: Calculate the curved surface area (CSA)
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi r l \]
Substituting the values we have:
\[ \text{CSA} = \frac{22}{7} \times 7 \times 25 \]
The \( 7 \) cancels out:
\[ \text{CSA} = 22 \times 25 \]
Calculating this gives:
\[ \text{CSA} = 550 \text{ cm}^2 \]

### Final Answer
The curved surface area of the cone is **550 cm²**.