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 total surface area (in `cm^(2)`)

Options

A. 2200
B. 1100
C. 2354
D. 2816

Answer

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

### Step 1: Find the radius of the base of the cone.
The perimeter (circumference) of the base of the cone is given as 88 cm. The formula for the circumference of a circle is:

\[ C = 2\pi r \]

We can rearrange this to solve for the radius \( r \):

\[ r = \frac{C}{2\pi} \]

Substituting the value of \( C \):

\[ r = \frac{88}{2 \times \frac{22}{7}} = \frac{88 \times 7}{44} = 14 \text{ cm} \]

### Step 2: Find the slant height of the cone.
The slant height \( l \) of the cone can be found using the Pythagorean theorem, which states:

\[ l = \sqrt{h^2 + r^2} \]

Where \( h \) is the height of the cone (48 cm) and \( r \) is the radius we just calculated (14 cm):

\[ l = \sqrt{48^2 + 14^2} = \sqrt{2304 + 196} = \sqrt{2500} = 50 \text{ cm} \]

### Step 3: Calculate the total surface area of the cone.
The total surface area \( A \) of a cone is given by the formula:

\[ A = \pi r l + \pi r^2 \]

We can factor out \( \pi r \):

\[ A = \pi r (l + r) \]

Substituting the values we have:

\[ A = \frac{22}{7} \times 14 \times (50 + 14) \]

Calculating \( l + r \):

\[ l + r = 50 + 14 = 64 \]

Now substituting this back into the area formula:

\[ A = \frac{22}{7} \times 14 \times 64 \]

Calculating \( \frac{22}{7} \times 14 \):

\[ \frac{22 \times 14}{7} = 44 \]

Now multiply by 64:

\[ A = 44 \times 64 = 2816 \text{ cm}^2 \]

### Final Answer:
The total surface area of the cone is \( 2816 \text{ cm}^2 \).

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