Cylinder Volume Calculation

The volume of a cylinder is given by the formula $ V_{cylinder} = \pi r^2 h $, where $ r $ is the radius and $ h $ is the height (or length).

• Given cylinder radius $ r_{cyl} = 1.4 $ cm.
• Given cylinder length $ h_{cyl} = 24 $ cm.
• Calculate cylinder volume: $ V_{cyl} = \pi \times (1.4 \text{ cm})^2 \times (24 \text{ cm}) $ $ V_{cyl} = \pi \times 1.96 \text{ cm}^2 \times 24 \text{ cm} $ $ V_{cyl} = 47.04 \pi \text{ cm}^3 $

Spherical Ball Volume Calculation

The volume of a sphere is given by the formula $ V_{sphere} = \frac{4}{3} \pi r^3 $, where $ r $ is the radius.

• Given spherical ball radius $ r_{sphere} = 2 $ mm.
• Convert radius to cm: $ r_{sphere} = 2 \text{ mm} = 0.2 \text{ cm} $.
• Calculate the volume of one spherical ball: $ V_{sphere} = \frac{4}{3} \pi \times (0.2 \text{ cm})^3 $ $ V_{sphere} = \frac{4}{3} \pi \times 0.008 \text{ cm}^3 $ $ V_{sphere} = \frac{0.032}{3} \pi \text{ cm}^3 $

Maximum Balls Calculation

To find the maximum number of balls, divide the total volume of the cylinder by the volume of a single spherical ball.

• Number of balls $ N = \frac{V_{cyl}}{V_{sphere}} $
• Substitute the calculated volumes: $ N = \frac{47.04 \pi \text{ cm}^3}{\frac{0.032}{3} \pi \text{ cm}^3} $
• Simplify the expression: $ N = \frac{47.04}{\frac{0.032}{3}} $ $ N = 47.04 \times \frac{3}{0.032} $ $ N = \frac{141.12}{0.032} $ $ N = 4410 $

Therefore, 4410 maximum spherical balls could be made.