If s denotes the area of the curved surface of a right circular cone of height h and semivertical angle `alpha` then S equals

Options

A. a)`pi h^(2) tan^(2) alpha`
B. b)`(1)/(3) pi h^(2) tan^(2) alpha`
C. c)`pi h^(2) sec alpha tan alpha`
D. d)`(1)/(3) pi h^(2) sec alpha tan alpha`

Answer

To find the area of the curved surface of a right circular cone given its height \( h \) and semi-vertical angle \( \alpha \), we can follow these steps:

### Step 1: Understand the Geometry of the Cone
We have a right circular cone with:
- Height \( h \)
- Radius \( r \)
- Slant height \( l \)
- Semi-vertical angle \( \alpha \)

### Step 2: Relate the Radius and Height using the Semi-Vertical Angle
From the definition of tangent in a right triangle, we can express the radius \( r \) in terms of height \( h \) and the semi-vertical angle \( \alpha \):
\[ \tan(\alpha) = \frac{r}{h} \]
From this, we can solve for \( r \):
\[ r = h \tan(\alpha) \]

### Step 3: Calculate the Slant Height
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the expression for \( r \):
\[ l = \sqrt{h^2 + (h \tan(\alpha))^2} \]
This simplifies to:
\[ l = \sqrt{h^2 + h^2 \tan^2(\alpha)} = \sqrt{h^2(1 + \tan^2(\alpha))} \]
Using the identity \( 1 + \tan^2(\alpha) = \sec^2(\alpha) \):
\[ l = h \sec(\alpha) \]

### Step 4: Calculate the Curved Surface Area
The formula for the curved surface area \( S \) of a cone is given by:
\[ S = \pi r l \]
Substituting the expressions for \( r \) and \( l \):
\[ S = \pi (h \tan(\alpha))(h \sec(\alpha)) \]
This simplifies to:
\[ S = \pi h^2 \tan(\alpha) \sec(\alpha) \]

### Final Result
Thus, the area of the curved surface of the cone is:
\[ S = \pi h^2 \tan(\alpha) \sec(\alpha) \]