The height to which a cylindrical vessel be filled with a homogenous liquid, to make the average force with which the liquid presses the side of the vessel equal to the force exerted by the liquid on the bottom of the vessel, is equal to.

Options

A. half of the radius of the vessel
B. radius of the vessel
C. one-fourth of the radius of the vessel
D. three-fourth of the radius of the vessel.

Explanation

To solve the problem, we need to find the height \( h \) of a homogeneous liquid in a cylindrical vessel such that the average force exerted by the liquid on the sides of the vessel equals the force exerted by the liquid on the bottom of the vessel.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The force exerted by the liquid on the bottom of the vessel can be calculated using the formula:
\[ F_{\text{bottom}} = P_{\text{bottom}} \times A \]
where \( P_{\text{bottom}} = \rho g h \) (pressure at the bottom), \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( A \) is the area of the base of the cylinder.

2. **Calculating the Force on the Bottom**:
- Therefore, we have:
\[ F_{\text{bottom}} = (\rho g h) \times A \]

3. **Calculating the Average Force on the Sides**:
- The pressure at any height \( y \) in the liquid is given by \( P(y) = \rho g y \).
- The average pressure on the side of the vessel from height 0 to height \( h \) is:
\[ P_{\text{avg}} = \frac{P_{\text{min}} + P_{\text{max}}}{2} = \frac{0 + \rho g h}{2} = \frac{\rho g h}{2} \]
- The force exerted by this average pressure on the side of the vessel is:
\[ F_{\text{sides}} = P_{\text{avg}} \times \text{Area of the side} \]
- The area of the side of the cylinder is \( 2\pi r h \), where \( r \) is the radius of the cylinder.

4. **Calculating the Force on the Sides**:
- Thus, we have:
\[ F_{\text{sides}} = \left(\frac{\rho g h}{2}\right) \times (2\pi r h) = \rho g h \pi r h \]

5. **Setting the Forces Equal**:
- To find the height \( h \) where the average force on the sides equals the force on the bottom, we set:
\[ F_{\text{bottom}} = F_{\text{sides}} \]
- This gives us:
\[ (\rho g h) A = \rho g h \pi r h \]

6. **Simplifying the Equation**:
- Since \( A = \pi r^2 \) (the area of the base), we substitute:
\[ (\rho g h) (\pi r^2) = \rho g h \pi r h \]
- Dividing both sides by \( \rho g h \) (assuming \( h \neq 0 \)):
\[ \pi r^2 = \pi r h \]

7. **Solving for Height**:
- Dividing both sides by \( \pi r \) (assuming \( r \neq 0 \)):
\[ r = h \]
- Therefore, the height \( h \) to which the cylindrical vessel should be filled is equal to the radius \( r \) of the cylinder.

### Final Answer:
\[ h = r \]