A cylindrical vessel, whose diameter and height both are equal to 30 cm is placed on a horizontal surface and a small particle P is placed in it at the distance of 5 cm from center. An eye is placed at a position such that the edge of the bottom is just visible.The particle P is in the plane of drawing.Up to what min. height should the water be poured in the vessel to make the particle P visible ??

To determine the minimum height of water needed in the cylindrical vessel for the particle P to be visible, we can use some principles of geometry and optics. The key here is to understand the relationship between the height of the water, the position of the particle, and the line of sight from the observer's eye to the particle.

Understanding the Geometry of the Situation

First, let's visualize the setup. We have a cylindrical vessel with a diameter of 30 cm, which means its radius is 15 cm (since radius = diameter/2). The height of the cylinder is also 30 cm. The particle P is located 5 cm from the center of the base of the cylinder, which places it at a horizontal distance of 5 cm from the center line of the cylinder.

Position of the Eye

The observer's eye is positioned such that the edge of the bottom of the cylinder is just visible. This means that the line of sight from the observer's eye to the particle P must not be obstructed by the water. The observer's eye is effectively at the height of the bottom edge of the cylinder, which is 0 cm in height.

Calculating the Required Height of Water

To find the minimum height of water required to make the particle visible, we can use the concept of similar triangles. The line of sight from the observer's eye to the particle P creates a triangle with the following points:

• Point A: The observer's eye at (0, 0)
• Point B: The position of the particle P at (5, h), where h is the height of the water
• Point C: The edge of the cylinder at (15, 0)

Using the properties of similar triangles, we can set up a proportion based on the distances:

Setting Up the Proportion

The horizontal distance from the center to the edge of the cylinder is 15 cm, and the horizontal distance from the center to the particle P is 5 cm. The height of the water (h) will create a triangle with the following ratio:

Height of water (h) / Horizontal distance to particle (5 cm) = Height of cylinder (30 cm) / Horizontal distance to edge (15 cm)

Formulating the Equation

We can express this as:

h / 5 = 30 / 15

Simplifying this gives:

h / 5 = 2

Now, solving for h:

h = 2 * 5 = 10 cm

Conclusion

Therefore, the minimum height of water that needs to be poured into the cylindrical vessel for the particle P to be visible is 10 cm. This ensures that the line of sight from the observer's eye to the particle is unobstructed by the water, allowing for clear visibility of the particle at that height.