An invered conical shaped vessel is filled with water to its brim. The height of the vessel is 8 cm and radius of the open end is 5 cm. When a few solid spherical metallic balls each of radius `(1)/(2) cm` are dropped in the vessel, 25% water is overflowd. The number of balls is :

Options

A. 100
B. 400
C. 200
D. 150

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Calculate the volume of the conical vessel.
The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone.

Given:
- Height \( h = 8 \) cm
- Radius \( r = 5 \) cm

Substituting the values into the formula:

\[ V = \frac{1}{3} \pi (5)^2 (8) = \frac{1}{3} \pi (25)(8) = \frac{200}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of water that overflows.
Since 25% of the water overflows when the balls are dropped into the vessel, we need to calculate 25% of the total volume of the cone.

\[ \text{Volume of water overflowed} = 0.25 \times V = 0.25 \times \frac{200}{3} \pi = \frac{50}{3} \pi \, \text{cm}^3 \]

### Step 3: Calculate the volume of one metallic ball.
The volume \( V_b \) of a sphere is given by:

\[ V_b = \frac{4}{3} \pi r^3 \]

Given the radius of each ball \( r = \frac{1}{2} \) cm, we substitute this value into the formula:

\[ V_b = \frac{4}{3} \pi \left(\frac{1}{2}\right)^3 = \frac{4}{3} \pi \left(\frac{1}{8}\right) = \frac{4}{24} \pi = \frac{1}{6} \pi \, \text{cm}^3 \]

### Step 4: Set up the equation to find the number of balls.
Let \( n \) be the number of balls. The total volume of \( n \) balls is:

\[ \text{Total volume of balls} = n \times V_b = n \times \frac{1}{6} \pi \]

Since the total volume of water overflowed is equal to the volume of the balls, we can set up the equation:

\[ n \times \frac{1}{6} \pi = \frac{50}{3} \pi \]

### Step 5: Solve for \( n \).
We can cancel \( \pi \) from both sides:

\[ n \times \frac{1}{6} = \frac{50}{3} \]

Multiplying both sides by 6:

\[ n = 6 \times \frac{50}{3} = \frac{300}{3} = 100 \]

### Final Answer:
The number of balls is \( n = 100 \).

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