The perimeter of base of a right circular cone is 88 cm. If the height of the cone is 48 cm, then what is the curved surface area (in `cm^(2)`) of the cone ?

Options

A. 550
B. 1100
C. 2200
D. 2700

Answer

To find the curved surface area of a right circular cone given the perimeter of its base and its height, we can follow these steps:

### Step 1: Understand the given information
We are given:
- The perimeter (circumference) of the base of the cone = 88 cm
- The height of the cone (h) = 48 cm

### Step 2: Relate the perimeter to the radius
The perimeter of the base of a cone, which is circular, is given by the formula:
\[ \text{Perimeter} = 2\pi r \]
Substituting the given perimeter:
\[ 2\pi r = 88 \]
Using \(\pi \approx \frac{22}{7}\):
\[ 2 \times \frac{22}{7} \times r = 88 \]

### Step 3: Solve for the radius (r)
Now, we can simplify the equation:
\[ \frac{44}{7} r = 88 \]
To isolate \(r\), multiply both sides by \(\frac{7}{44}\):
\[ r = 88 \times \frac{7}{44} \]
Calculating:
\[ r = 88 \times \frac{1}{2} = 44 \text{ cm} \]
Now, divide by 2:
\[ r = 14 \text{ cm} \]

### 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 values:
\[ l = \sqrt{48^2 + 14^2} \]
Calculating \(48^2\) and \(14^2\):
\[ l = \sqrt{2304 + 196} = \sqrt{2500} \]
Thus,
\[ l = 50 \text{ cm} \]

### Step 5: Calculate the curved surface area (CSA)
The formula for the curved surface area (CSA) of a cone is:
\[ \text{CSA} = \pi r l \]
Substituting the values of \(r\) and \(l\):
\[ \text{CSA} = \frac{22}{7} \times 14 \times 50 \]
Calculating:
\[ \text{CSA} = \frac{22 \times 14 \times 50}{7} \]
First, simplify \(\frac{22 \times 14}{7}\):
\[ = 44 \times 50 = 2200 \text{ cm}^2 \]

### Final Answer
The curved surface area of the cone is \(2200 \text{ cm}^2\).

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