A container in the shape of a right circukar cone, whose radius and depth are equal, gets completely filled by 128000 sperical droplets, each of diameter 2mm. What is the radius (in cm) of the container?

Options

A. 6
B. 4
C. 8
D. 2

Answer

To solve the problem, we need to find the radius of a right circular cone that is filled with 128,000 spherical droplets, each with a diameter of 2 mm. Let's break down the solution step by step.

### Step 1: Find the radius of a single droplet
The diameter of each droplet is given as 2 mm. Therefore, the radius \( r_d \) of a single droplet is:
\[ r_d = \frac{\text{diameter}}{2} = \frac{2 \text{ mm}}{2} = 1 \text{ mm} \]
To convert this to centimeters:
\[ r_d = 1 \text{ mm} = 0.1 \text{ cm} \]

### Step 2: Calculate the volume of a single droplet
The volume \( V_d \) of a sphere is given by the formula:
\[ V_d = \frac{4}{3} \pi r_d^3 \]
Substituting \( r_d = 0.1 \text{ cm} \):
\[ V_d = \frac{4}{3} \pi (0.1)^3 = \frac{4}{3} \pi (0.001) = \frac{4\pi}{3000} \text{ cm}^3 \]

### Step 3: Calculate the total volume of all droplets
Since there are 128,000 droplets, the total volume \( V_{total} \) of the droplets is:
\[ V_{total} = 128000 \times V_d = 128000 \times \frac{4\pi}{3000} = \frac{128000 \times 4\pi}{3000} \text{ cm}^3 \]

### Step 4: Simplify the total volume
Calculating the total volume:
\[ V_{total} = \frac{128000 \times 4\pi}{3000} = \frac{512000\pi}{3000} \text{ cm}^3 \]

### Step 5: Set up the volume of the cone
The volume \( V_c \) of a right circular cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
Given that the radius \( r \) and height \( h \) of the cone are equal, we can set \( h = r \):
\[ V_c = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \]

### Step 6: Equate the volumes
Since the volume of the cone is equal to the total volume of the droplets:
\[ \frac{1}{3} \pi r^3 = \frac{512000\pi}{3000} \]

### Step 7: Cancel \( \pi \) and solve for \( r^3 \)
Cancelling \( \pi \) from both sides:
\[ \frac{1}{3} r^3 = \frac{512000}{3000} \]
Multiplying both sides by 3:
\[ r^3 = \frac{512000 \times 3}{3000} = \frac{1536000}{3000} = 512 \]

### Step 8: Find the radius \( r \)
Taking the cube root:
\[ r = \sqrt[3]{512} = 8 \text{ cm} \]

### Final Answer
The radius of the container is \( \boxed{8} \) cm.