Problem:
An iron sphere of radius 'a' is immersed completely in water contained in a right circular cone of semivertical angle 30 degrees. The water is drained off from the cone until it touches the surface of the cone. Find the volume of water remaining in the cone.

Solution:

To solve this problem, we need to find the volume of water remaining in the cone after the sphere is immersed and the water is drained off. Let's break the solution into smaller steps:

Step 1: Finding the height of the cone

Let h be the height of the cone. Since the cone touches the surface of the water, the height of the cone will be equal to the radius of the sphere (h = a).

Step 2: Finding the volume of the cone

The volume of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

Given that the semivertical angle of the cone is 30 degrees, the radius of the base can be found using the formula: r = h * tan(30°).

Substituting the values, we get: r = a * tan(30°).

Now, we can calculate the volume of the cone using the formula: V = (1/3) * π * (a * tan(30°))^2 * a.

Step 3: Finding the volume of the sphere

The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3, where r is the radius of the sphere.

Given that the radius of the sphere is 'a', we can calculate the volume of the sphere using the formula: V = (4/3) * π * a^3.

Step 4: Finding the volume of water remaining

The volume of water remaining in the cone after the sphere is immersed and the water is drained off can be calculated by subtracting the volume of the sphere from the volume of the cone.

Let V_water be the volume of water remaining. We can calculate it using the formula: V_water = V_cone - V_sphere.

Substituting the values, we get: V_water = (1/3) * π * (a * tan(30°))^2 * a - (4/3) * π * a^3.

Simplifying the equation, we get: V_water = (1/3) * π * a^3 * tan^2(30°) - (4/3) * π * a^3.

Final Answer:

The volume of water remaining in the cone is (1/3) * π * a^3 * (tan^2(30°) - 4).

Answer
5πa3/3