Find the total surface area (in `cm^2`) of a right circular cone of diameter 21 cm and slant height 11 cm.

Options

A. 467.5
B. 384
C. 724
D. 709.5

Answer

To find the total surface area of a right circular cone given the diameter and slant height, follow these steps:

### Step 1: Identify the given values
- Diameter (d) = 21 cm
- Slant height (l) = 11 cm

### Step 2: Calculate the radius
The radius (r) is half of the diameter.
\[ r = \frac{d}{2} = \frac{21}{2} = 10.5 \text{ cm} \]

### Step 3: Write the formula for the total surface area (TSA) of a cone
The formula for the total surface area of a right circular cone is:
\[ \text{TSA} = \pi r l + \pi r^2 \]
where:
- \( r \) = radius
- \( l \) = slant height

### Step 4: Substitute the values into the formula
Using \( \pi \approx \frac{22}{7} \):
\[ \text{TSA} = \frac{22}{7} \times r \times l + \frac{22}{7} \times r^2 \]
Substituting \( r = 10.5 \) cm and \( l = 11 \) cm:
\[ \text{TSA} = \frac{22}{7} \times 10.5 \times 11 + \frac{22}{7} \times (10.5)^2 \]

### Step 5: Calculate each part of the formula
1. Calculate \( \frac{22}{7} \times 10.5 \times 11 \):
\[ \frac{22}{7} \times 10.5 \times 11 = \frac{22 \times 10.5 \times 11}{7} = \frac{2521}{7} = 360.14 \text{ cm}^2 \]

2. Calculate \( \frac{22}{7} \times (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
\[ \frac{22}{7} \times 110.25 = \frac{2425.5}{7} = 346.5 \text{ cm}^2 \]

### Step 6: Add both parts to find the total surface area
\[ \text{TSA} = 360.14 + 346.5 = 706.64 \text{ cm}^2 \]

### Step 7: Round to the nearest decimal if necessary
The total surface area is approximately:
\[ \text{TSA} \approx 709.5 \text{ cm}^2 \]

### Final Answer
The total surface area of the right circular cone is \( 709.5 \text{ cm}^2 \).

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