Question

Three tennis balls are just packed in a cylindrical jar. If radius of each ball is r, volume of air inside the jar is

• A. \(2\pi r^3\)
• B. \(3\pi r^3\)
• C. \(5\pi r^3\)
• D. \(4\pi r^3\)

Hint:

According to a famous geometric theorem by Archimedes, the volume of a sphere is exactly \(\frac{2}{3}\) of the volume of its circumscribed cylinder.
This means the three balls occupy exactly \(\frac{2}{3}\) of the jar's volume, leaving \(\frac{1}{3}\) of the jar's volume as air space:
\[ V_{\text{air}} = \frac{1}{3} \times V_{\text{cylinder}} = \frac{1}{3} \times 6\pi r^3 = 2\pi r^3 \] Remembering this ratio saves you from doing long calculations!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Surface Areas and Volumes.
The tennis balls are spherical in shape, and they are packed tightly inside a cylindrical container.
Because the packing is tight, the dimensions of the cylinder are directly determined by the dimensions of the spherical balls.
The volume of air inside the jar is the empty space, which is found by subtracting the volume of the three tennis balls from the total volume of the cylindrical jar.

Step 2: Key Formula or Approach:
- Volume of a single sphere of radius \(r\):
\[ V_{\text{sphere}} = \frac{4}{3}\pi r^3 \] - Volume of a cylinder of base radius \(R\) and height \(H\):
\[ V_{\text{cylinder}} = \pi R^2 H \] - The volume of air inside the jar is calculated as:
\[ V_{\text{air}} = V_{\text{cylinder}} - 3 \times V_{\text{sphere}} \]

Step 3: Detailed Explanation:

• Determine the dimensions of the cylindrical jar:
- The base radius \(R\) of the cylinder is equal to the radius of one tennis ball: \(R = r\).
- Three tennis balls are stacked vertically inside the jar.
The height \(H\) of the cylinder is equal to three times the diameter of one tennis ball:
\[ H = 3 \times (2r) = 6r \]

• Calculate the total volume of the cylindrical jar:
\[ V_{\text{cylinder}} = \pi R^2 H \] \[ V_{\text{cylinder}} = \pi (r)^2 (6r) = 6\pi r^3 \]

• Calculate the total volume occupied by the three spherical tennis balls:
\[ \text{Volume of one ball} = \frac{4}{3}\pi r^3 \] \[ \text{Volume of three balls} = 3 \times \frac{4}{3}\pi r^3 = 4\pi r^3 \]

• Subtract the volume of the balls from the volume of the cylinder to find the volume of the air inside:
\[ V_{\text{air}} = V_{\text{cylinder}} - \text{Volume of three balls} \] \[ V_{\text{air}} = 6\pi r^3 - 4\pi r^3 = 2\pi r^3 \]

Step 4: Final Answer:
The volume of air inside the cylindrical jar is \(2\pi r^3\).
Therefore, the correct option is (A).