A cone full of water, is placed on its side on a horizontal table, the thrust on its base is x times the weight of the contained fluid, where `2alpha` is the vertical angle of the cone. Find the value of x.

Options

A. `3 cos alpha `
B. ` 3 sin alpha `
C. `2 sin alpha `
D. ` 2 cos alpha `

Answer

To solve the problem, we need to determine the value of \( x \) such that the thrust on the base of a cone full of water placed on its side is \( x \) times the weight of the contained fluid.

### Step-by-Step Solution:

1. **Understanding the Geometry of the Cone**:
- The cone is placed on its side, and we denote the vertical angle of the cone as \( 2\alpha \). Thus, the half-angle at the base of the cone is \( \alpha \).
- The radius \( r \) of the base of the cone can be related to the height \( h \) of the cone using the tangent function:
\[ r = h \tan \alpha \]

2. **Calculating the Volume of the Cone**:
- The volume \( V \) of the cone can be expressed as:
\[ V = \frac{1}{3} \pi r^2 h \]
- Substituting \( r = h \tan \alpha \):
\[ V = \frac{1}{3} \pi (h \tan \alpha)^2 h = \frac{1}{3} \pi h^3 \tan^2 \alpha \]

3. **Calculating the Weight of the Fluid**:
- The weight \( W \) of the fluid contained in the cone is given by:
\[ W = \rho V g = \rho \left(\frac{1}{3} \pi h^3 \tan^2 \alpha\right) g \]
- Therefore, the weight of the fluid becomes:
\[ W = \frac{1}{3} \pi \rho g h^3 \tan^2 \alpha \]

4. **Calculating the Thrust on the Base**:
- The thrust \( T \) on the base of the cone is given by the product of pressure and area. The pressure at the base due to the height of the fluid is:
\[ P = \rho g h \]
- The area \( A \) of the base is:
\[ A = \pi r^2 = \pi (h \tan \alpha)^2 = \pi h^2 \tan^2 \alpha \]
- Thus, the thrust \( T \) can be calculated as:
\[ T = P \cdot A = (\rho g h) \cdot (\pi h^2 \tan^2 \alpha) = \pi \rho g h^3 \tan^2 \alpha \]

5. **Relating Thrust to Weight**:
- We need to find \( x \) such that:
\[ T = x W \]
- Substituting the expressions for \( T \) and \( W \):
\[ \pi \rho g h^3 \tan^2 \alpha = x \left(\frac{1}{3} \pi \rho g h^3 \tan^2 \alpha\right) \]
- Simplifying this equation:
\[ 1 = \frac{x}{3} \]
- Therefore, we find:
\[ x = 3 \]

### Final Answer:
The value of \( x \) is \( 3 \).