Tennis balls withal diameter of 2.5 in. Are sold in cans of three. The can is a cylinder. What is the volume of the space in the can not occupied by tennis balls? Assume the balls touch the on the sides,top and bottom.

Height of cylinder, h

= 3 times the diameter of tennis balls
= 3*2.5"
= 7.5"
Diameter of cylinder, d
= diameter of tennis ball
= 2.5"
Volume of cylinder
= πd²h/4

Volume of each tennis ball, v
= 4π(d/2)³/3
Volume of three tennis balls
= 3v
= 4π(d/2)³

Interstitial space
= Volume of cylinder - volume of three balls.

approximately 12.27 in^3

To find the volume of the space in the can not occupied by tennis balls, we first need to find the volume of the can.

The can is in the shape of a cylinder, and we need to know the dimensions of the can in order to calculate its volume.

Let's assume that the can has a height (h) of 5 inches and a diameter (d) of 3 inches.

The formula to calculate the volume of a cylinder is V = πr^2h, where r is the radius of the cylinder (which is half of the diameter).

Given that the diameter is 3 inches, the radius (r) would be 1.5 inches.

Plugging the values into the formula, we get:

V = π(1.5^2)(5)

Calculating this, we find:

V = π(2.25)(5)
V = 11.25π cubic inches

So, the volume of the can is 11.25π cubic inches.

Now, let's find the volume occupied by the tennis balls.

Since the balls touch each other and the sides, top, and bottom, they form a close-packed arrangement, known as the hexagonal close-packed (HCP) structure.

In this arrangement, the packing efficiency is approximately 74%.

Given that the diameter of the tennis balls is 2.5 inches, the radius (r) would be 1.25 inches.

The formula to calculate the volume of a sphere is V = (4/3)πr^3.

Plugging in the values, we can calculate the individual volume of one tennis ball:

V_ball = (4/3)π(1.25)^3
V_ball = (4/3)π(1.953125)
V_ball ≈ 8.1676 cubic inches

Since there are 3 tennis balls in a can, the total volume occupied by the tennis balls is:

V_balls = 3V_ball
V_balls = 3(8.1676)
V_balls ≈ 24.5028 cubic inches

Finally, to find the volume of the space in the can not occupied by tennis balls, we subtract the volume occupied by the tennis balls from the volume of the can:

V_space = V_can - V_balls
V_space = 11.25π - 24.5028

Hence, the volume of the space in the can not occupied by tennis balls is 11.25π - 24.5028 cubic inches.

To find the volume of the space in the can not occupied by tennis balls, we can first calculate the volume of the entire can and then subtract the volume of the tennis balls.

1. Calculate the volume of the entire can:
Since the can is in the shape of a cylinder, the volume formula for a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.

Given:
- Diameter of the tennis balls = 2.5 inches
- Radius of the tennis balls = diameter / 2 = 2.5 / 2 = 1.25 inches

Let's assume the height of the can is h inches (this is not provided in the question). We'll use this variable for now.

So, the volume of the entire can = π * r^2 * h = π * 1.25^2 * h = 1.25^2 * π * h = 1.5625 * π * h.

2. Calculate the volume of the tennis balls:
Since the tennis balls touch each other on the sides, top, and bottom, the volume of the tennis balls can be calculated as if they were packed tightly in a rectangular prism.

Given:
- Diameter of the tennis balls = 2.5 inches
- Radius of the tennis balls = diameter / 2 = 2.5 / 2 = 1.25 inches

Each tennis ball can be considered as a sphere, and the formula to calculate the volume of a sphere is V = (4/3) * π * r^3.

Since there are three tennis balls in the can, the total volume occupied by the tennis balls is (4/3) * π * r^3 * 3 = 4 * π * r^3.

3. Calculate the volume of the space not occupied by the tennis balls:
To get the volume of the space not occupied by the tennis balls, subtract the volume of the tennis balls from the volume of the entire can:

Volume of the space not occupied = Volume of the entire can - Volume of the tennis balls
= 1.5625 * π * h - 4 * π * r^3.

Simplifying, we get:
Volume of the space not occupied = π * (1.5625h - 4r^3).

Now you can plug in the known values for the height of the can and the radius of the tennis balls to calculate the volume of the space not occupied by the tennis balls.