If from a circular sheet of paper of radius 15 c, a sector of `144^(@)` is removed and the remaining is used to make a conical surface, then the angle at the vertex will be :

Options

A. `sin^(-1)((3)/(10))`
B. `sin^(-1)((6)/(5))`
C. ` 2sin^(-1)((3)/(5))`
D. `2 sin^(-1)((4)/(5))`

Answer

To solve the problem, we need to determine the angle at the vertex of the cone formed by the remaining sector of the circular sheet after removing a sector of 144°.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of the circular sheet (R) = 15 cm
- Angle of the removed sector = 144°

2. **Calculate the Angle of the Remaining Sector:**
- The total angle of a circle is 360°.
- Angle of the remaining sector = 360° - 144° = 216°.

3. **Calculate the Arc Length of the Remaining Sector:**
- The formula for the arc length (L) of a sector is given by:
\[ L = \frac{\theta}{360} \times 2\pi R \]
- Here, θ = 216° and R = 15 cm.
- Substituting the values:
\[ L = \frac{216}{360} \times 2\pi \times 15 \]
- Simplifying:
\[ L = \frac{3}{5} \times 2\pi \times 15 = \frac{6\pi \times 15}{5} = 18\pi \text{ cm} \]

4. **Determine the Radius of the Base of the Cone:**
- The arc length of the remaining sector becomes the circumference of the base of the cone.
- The circumference (C) of the base of the cone is given by:
\[ C = 2\pi r \]
- Setting the two equal:
\[ 2\pi r = 18\pi \]
- Dividing both sides by \(2\pi\):
\[ r = \frac{18\pi}{2\pi} = 9 \text{ cm} \]

5. **Determine the Slant Height of the Cone:**
- The slant height (l) of the cone is equal to the radius of the original circular sheet:
\[ l = 15 \text{ cm} \]

6. **Use the Sine Function to Find the Vertex Angle:**
- In the right triangle formed by the radius (r), slant height (l), and the height (h) of the cone:
- We know:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{r}{l} = \frac{9}{15} = \frac{3}{5} \]
- Therefore:
\[ \theta = \sin^{-1}\left(\frac{3}{5}\right) \]

7. **Calculate the Vertex Angle:**
- The vertex angle (V) of the cone is twice the angle θ:
\[ V = 2\theta = 2 \sin^{-1}\left(\frac{3}{5}\right) \]

### Final Answer:
The angle at the vertex of the cone is:
\[ \boxed{2 \sin^{-1}\left(\frac{3}{5}\right)} \]