A cone is made of a sector with a radius of 14 cm and an angle of ` 60^(@)`. What is total surface area of the cone ?

Options

A. `119.78 cm^(2) `
B. `191.87 cm^(2) `
C. `196.5 cm^(2) `
D. none of these

Answer

To find the total surface area of a cone made from a sector with a radius of 14 cm and an angle of 60 degrees, follow these steps:

### Step 1: Find the Slant Height of the Cone
The slant height (l) of the cone is equal to the radius of the sector. Therefore,
\[ l = 14 \, \text{cm} \]

### Step 2: Calculate the Circumference of the Base of the Cone
The circumference of the base of the cone is equal to the length of the arc of the sector. The formula for the arc length (circumference of the base) is:
\[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \]
Where:
- \( \theta = 60^\circ \)
- \( r = 14 \, \text{cm} \)

Substituting the values:
\[ \text{Arc Length} = \frac{60}{360} \times 2\pi \times 14 \]
\[ = \frac{1}{6} \times 2\pi \times 14 \]
\[ = \frac{28\pi}{6} \]
\[ = \frac{14\pi}{3} \, \text{cm} \]

### Step 3: Find the Radius of the Base of the Cone
The circumference of the base of the cone is given by:
\[ C = 2\pi r \]
Setting this equal to the arc length:
\[ 2\pi r = \frac{14\pi}{3} \]

To find \( r \):
\[ r = \frac{14\pi}{3 \times 2\pi} \]
\[ = \frac{14}{6} \]
\[ = \frac{7}{3} \, \text{cm} \]

### Step 4: Calculate the Total Surface Area of the Cone
The total surface area (TSA) of a cone is given by:
\[ \text{TSA} = \pi r^2 + \pi r l \]
Where:
- \( r = \frac{7}{3} \, \text{cm} \)
- \( l = 14 \, \text{cm} \)

Calculating \( \pi r^2 \):
\[ \pi r^2 = \pi \left(\frac{7}{3}\right)^2 = \pi \left(\frac{49}{9}\right) = \frac{49\pi}{9} \]

Calculating \( \pi r l \):
\[ \pi r l = \pi \left(\frac{7}{3}\right) \times 14 = \frac{98\pi}{3} \]

Now, combine both parts:
\[ \text{TSA} = \frac{49\pi}{9} + \frac{98\pi}{3} \]

To add these fractions, convert \( \frac{98\pi}{3} \) to have a common denominator of 9:
\[ \frac{98\pi}{3} = \frac{294\pi}{9} \]

Now, add:
\[ \text{TSA} = \frac{49\pi}{9} + \frac{294\pi}{9} = \frac{343\pi}{9} \]

### Step 5: Substitute \( \pi \) and Calculate
Using \( \pi \approx \frac{22}{7} \):
\[ \text{TSA} = \frac{343 \times \frac{22}{7}}{9} \]
\[ = \frac{343 \times 22}{63} \]
\[ = \frac{7566}{63} \]
\[ \approx 119.78 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cone is approximately:
\[ \text{TSA} \approx 119.78 \, \text{cm}^2 \]

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