A cylindrical box of radius 5 cm contains 10 solid spherical balls, each of radius 5 cm. If the top most ball touches the upper cover of the box, then volume of the empty space in the box is

Options

A. `(2500)/(3) pi cm^(3) `
B. `5000 pi cm^(3) `
C. `2500 pi cm^(3) `
D. `(5000)/(3) pi cm^(3) `

Answer

To find the volume of the empty space in the cylindrical box containing 10 solid spherical balls, we will follow these steps:

### Step 1: Calculate the volume of the cylindrical box.
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius of the cylindrical box \( r = 5 \) cm
- Since the topmost ball touches the upper cover of the box and each ball has a radius of 5 cm, the height of the box can be calculated as follows:
- The height of one ball (diameter) = \( 2 \times 5 \) cm = 10 cm
- Total height of 10 balls = \( 10 \times 10 \) cm = 100 cm

Now substituting the values into the volume formula:
\[ V_{\text{cylinder}} = \pi (5)^2 (100) = \pi (25)(100) = 2500\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one spherical ball.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Given the radius of each ball \( r = 5 \) cm, we can substitute this value into the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \text{ cm}^3 \]

### Step 3: Calculate the total volume of all 10 spherical balls.
To find the total volume of 10 balls, we multiply the volume of one ball by 10:
\[ V_{\text{total spheres}} = 10 \times V_{\text{sphere}} = 10 \times \frac{500}{3} \pi = \frac{5000}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the empty space in the box.
The volume of the empty space can be found by subtracting the total volume of the spheres from the volume of the cylinder:
\[ V_{\text{empty}} = V_{\text{cylinder}} - V_{\text{total spheres}} = 2500\pi - \frac{5000}{3}\pi \]

To perform this subtraction, we need a common denominator:
\[ V_{\text{empty}} = \frac{7500}{3}\pi - \frac{5000}{3}\pi = \frac{7500 - 5000}{3}\pi = \frac{2500}{3}\pi \text{ cm}^3 \]

### Final Answer:
The volume of the empty space in the box is:
\[ \frac{2500}{3}\pi \text{ cm}^3 \]

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