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**Problem 3:** (As interpreted from the image)

Given a cone with a slant height \( L \), and a sector with radius \( L \) that is rolled up to form the cone, the central angle of the sector is given as \( \theta \). The cone has a base radius \( R \) and height \( h \).

**a) Find the value of \( \theta \).**

**b) Show that \( \tan \alpha = \frac{R}{h} \) where \( \alpha \) is the semi-vertical angle of the cone.**

**c) Find the expression for the curved surface area of the cone.**

### Solution:

#### Part (a) Finding the value of \( \theta \):

The circumference of the base of the cone is equal to the arc length of the sector. The arc length \( S \) of a sector with radius \( L \) and central angle \( \theta \) (in radians) is given by:
\[
S = L \theta
\]
Since this arc length is equal to the circumference of the base of the cone \( 2\pi R \):
\[
L \theta = 2\pi R
\]
Thus, the value of \( \theta \) is:
\[
\theta = \frac{2\pi R}{L}
\]

#### Part (b) Showing that \( \tan \alpha = \frac{R}{h} \):

In a right-angled triangle formed by the height \( h \), the radius \( R \), and the slant height \( L \) of the cone:
\[
L^2 = R^2 + h^2
\]
The semi-vertical angle \( \alpha \) is the angle opposite to \( R \) and adjacent to \( h \). Hence, by definition of tangent:
\[
\tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{R}{h}
\]
This confirms that \( \tan \alpha = \frac{R}{h} \).

#### Part (c) Finding the expression for the curved surface area of the cone:

The curved surface area (CSA) of a cone is given by:
\[
\text{CSA} = \text{radius} \times \text{slant height} \times \theta
\]
Since the entire surface of the sector forms the curved surface of the cone:
\[
\text{CSA} = \frac{1}{2} \times L \times L \theta = \frac{1}{2} \times L^2 \times \frac{2\pi R}{L} = \pi R L
\]

The curved surface area of the cone is thus \( \pi R L \).

Would you like any further details or have any questions?

### Related Questions:
1. How is the concept of arc length applied in other geometric shapes like circles and ellipses?
2. Can you derive the formula for the volume of a cone using integration?
3. What are the conditions under which a sector can form a complete cone when rolled up?
4. How would the problem change if the cone was frustum-shaped instead of complete?
5. What is the significance of the semi-vertical angle in the design of conical structures?

**Tip:** When solving problems involving geometric shapes, always ensure that the relationships between the dimensions are consistent with the properties of the shape (e.g., Pythagoras' theorem in right triangles).

Explore how to solve a geometry problem involving a cone's slant height, sector angle, and base dimensions. This problem covers finding sector angle theta, verifying tangent relation in right triangles, and calculating curved surface area of a cone. Suitable for high school students learning geometry and trigonometry.

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