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

The given problem is about finding the volume of air inside a cylindrical jar when three tennis balls are packed in it. Let's solve this step-by-step:

Given: Each tennis ball has a radius \( r \).

The diameter of each tennis ball is \( 2r \). Therefore, the diameter of the cylindrical jar is also \( 2r \) (since the balls are just packed inside without gaps laterally), making the radius of the cylindrical jar itself equal to \( r \).

Height of the cylindrical jar: Since there are three tennis balls packed one above the other, the total height will be three times the diameter of each ball.

• Height, \( h = 3 \times 2r = 6r \).

Calculate the volume of the cylindrical jar:

• Formula for the volume of a cylinder: \(V_{\text{cylinder}} = \pi r^2 h\)
• Substitute the values: \(V_{\text{cylinder}} = \pi r^2 (6r) = 6\pi r^3\)

Calculate the total volume of the three tennis balls:

• Volume of one ball: \(V_{\text{ball}} = \frac{4}{3} \pi r^3\)
• Total volume of three balls: \(V_{\text{balls}} = 3 \times \frac{4}{3} \pi r^3 = 4\pi r^3\)

Calculate the volume of air inside the jar:

• Volume of air = Volume of the cylindrical jar - Volume of three balls.
• \(V_{\text{air}} = 6\pi r^3 - 4\pi r^3 = 2\pi r^3\)

Hence, the volume of air inside the jar is 2πr³.

Therefore, the correct answer is: \(2\pi r^3\).