A sector of a circle of radius 12 cm has the angle `120^(@)` It is rolled up so that two bounding radii are joined together to form a cone. Find the volume of the cone.

Options

A. `189.6cm^(3)`
B. `169.5cm^(3)`
C. `179.6cm^(3)`
D. `125.5cm^(3)`

Answer

To find the volume of the cone formed by rolling up a sector of a circle with a radius of 12 cm and an angle of 120 degrees, we can follow these steps:

### Step 1: Find the Length of the Arc of the Sector
The length of the arc (L) of the sector can be calculated using the formula:
\[ L = \frac{\theta}{360} \times 2\pi r \]
where:
- \( \theta = 120^\circ \)
- \( r = 12 \, \text{cm} \)

Substituting the values:
\[ L = \frac{120}{360} \times 2\pi \times 12 \]
\[ L = \frac{1}{3} \times 2\pi \times 12 \]
\[ L = 8\pi \, \text{cm} \]

### Step 2: Relate the Length of the Arc to the Circumference of the Base of the Cone
When the sector is rolled up to form a cone, the length of the arc becomes the circumference (C) of the base of the cone:
\[ C = L = 8\pi \, \text{cm} \]

The circumference of the base of the cone can also be expressed as:
\[ C = 2\pi r \]
where \( r \) is the radius of the base of the cone. Setting the two equations for circumference equal gives:
\[ 2\pi r = 8\pi \]

### Step 3: Solve for the Radius of the Base of the Cone
Dividing both sides by \( 2\pi \):
\[ r = \frac{8\pi}{2\pi} = 4 \, \text{cm} \]

### Step 4: Find the Height of the Cone
The radius of the original sector (which becomes the slant height \( l \) of the cone) is 12 cm. We can use the Pythagorean theorem to find the height \( h \) of the cone:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 12^2 = 4^2 + h^2 \]
\[ 144 = 16 + h^2 \]
\[ h^2 = 144 - 16 = 128 \]
\[ h = \sqrt{128} = 8\sqrt{2} \, \text{cm} \]

### Step 5: Calculate the Volume of the Cone
The volume \( V \) of the cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V = \frac{1}{3} \pi (4)^2 (8\sqrt{2}) \]
\[ V = \frac{1}{3} \pi (16)(8\sqrt{2}) \]
\[ V = \frac{128\sqrt{2}}{3} \pi \, \text{cm}^3 \]

### Final Answer
The volume of the cone is:
\[ V = \frac{128\sqrt{2}}{3} \pi \, \text{cm}^3 \]

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