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:

When spheres are "just packed" in a cylinder, the cylinder's height is always equal to the sum of the diameters of the spheres. For \(n\) spheres, \(h = n \times 2r\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The "volume of air" is the volume of the cylinder that is not occupied by the three spherical balls.
Step 2: Key Formula or Approach:
Radius of cylinder = Radius of ball = \(r\).
Height of cylinder (\(h\)) = \(3 \times\) Diameter of ball = \(3 \times (2r) = 6r\).
Volume of cylinder = \(\pi r^2 h\).
Volume of 3 spheres = \(3 \times \left( \frac{4}{3} \pi r^3 \right)\).
Step 3: Detailed Explanation:
Calculate volume of cylinder:
\[ V_{cyl} = \pi \times r^2 \times (6r) = 6 \pi r^3 \]
Calculate volume of 3 balls:
\[ V_{balls} = 3 \times \frac{4}{3} \pi r^3 = 4 \pi r^3 \]
Volume of air = \(V_{cyl} - V_{balls}\):
\[ V_{air} = 6 \pi r^3 - 4 \pi r^3 = 2 \pi r^3 \]
Step 4: Final Answer:
The volume of air inside the jar is \(2\pi r^3\).