Question
Tennis balls with a diameter of 6.6 cm are sold in cans of three. The can is a cylinder. What is the volume of the space NOT occupied by the tennis balls?
Answers
GPT-4o mini
To find the volume of the space not occupied by the tennis balls in the cylindrical can, we first need to calculate the volumes of the balls and the can itself.
### Step 1: Calculate the volume of one tennis ball
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the diameter of a tennis ball is 6.6 cm, the radius \( r \) is:
\[
r = \frac{6.6 \text{ cm}}{2} = 3.3 \text{ cm}
\]
Now, we can calculate the volume of one tennis ball:
\[
V_{\text{ball}} = \frac{4}{3} \pi (3.3)^3
\]
Calculating \( (3.3)^3 \):
\[
(3.3)^3 = 3.3 \times 3.3 \times 3.3 = 35.937
\]
Thus,
\[
V_{\text{ball}} = \frac{4}{3} \pi (35.937) \approx \frac{4}{3} \cdot 3.14159 \cdot 35.937 \approx 150.267 \text{ cm}^3
\]
Now, since there are three balls:
\[
V_{\text{total balls}} = 3 \times V_{\text{ball}} = 3 \times 150.267 \approx 450.801 \text{ cm}^3
\]
### Step 2: Calculate the volume of the cylindrical can
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius of the base and \( h \) is the height of the cylinder.
To determine the height of the cylinder, we note that three tennis balls are stacked in a can. The height of three tennis balls (each with a diameter of 6.6 cm) is:
\[
h = 3 \times 6.6 \text{ cm} = 19.8 \text{ cm}
\]
The radius \( r \) of the can must also be at least the radius of a ball to fit them properly (the exact dimensions may involve some extra space in the can, but we're simplifying here). So, the radius of the can is also approximately:
\[
r = 3.3 \text{ cm}
\]
Now we can calculate the volume of the can:
\[
V_{\text{can}} = \pi (3.3)^2 (19.8)
\]
Calculating \( (3.3)^2 \):
\[
(3.3)^2 = 10.89
\]
Thus:
\[
V_{\text{can}} \approx \pi (10.89)(19.8) \approx 3.14159 \times 215.562 \approx 678.014 \text{ cm}^3
\]
### Step 3: Calculate the space not occupied by the tennis balls
To find the volume of the space not occupied by the tennis balls, we subtract the volume of the balls from the volume of the can:
\[
V_{\text{unused}} = V_{\text{can}} - V_{\text{total balls}}
\]
Substituting the values we calculated:
\[
V_{\text{unused}} = 678.014 - 450.801 \approx 227.213 \text{ cm}^3
\]
### Final Answer
The volume of the space not occupied by the tennis balls is approximately:
\[
\boxed{227.21} \text{ cm}^3
\]
### Step 1: Calculate the volume of one tennis ball
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the diameter of a tennis ball is 6.6 cm, the radius \( r \) is:
\[
r = \frac{6.6 \text{ cm}}{2} = 3.3 \text{ cm}
\]
Now, we can calculate the volume of one tennis ball:
\[
V_{\text{ball}} = \frac{4}{3} \pi (3.3)^3
\]
Calculating \( (3.3)^3 \):
\[
(3.3)^3 = 3.3 \times 3.3 \times 3.3 = 35.937
\]
Thus,
\[
V_{\text{ball}} = \frac{4}{3} \pi (35.937) \approx \frac{4}{3} \cdot 3.14159 \cdot 35.937 \approx 150.267 \text{ cm}^3
\]
Now, since there are three balls:
\[
V_{\text{total balls}} = 3 \times V_{\text{ball}} = 3 \times 150.267 \approx 450.801 \text{ cm}^3
\]
### Step 2: Calculate the volume of the cylindrical can
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius of the base and \( h \) is the height of the cylinder.
To determine the height of the cylinder, we note that three tennis balls are stacked in a can. The height of three tennis balls (each with a diameter of 6.6 cm) is:
\[
h = 3 \times 6.6 \text{ cm} = 19.8 \text{ cm}
\]
The radius \( r \) of the can must also be at least the radius of a ball to fit them properly (the exact dimensions may involve some extra space in the can, but we're simplifying here). So, the radius of the can is also approximately:
\[
r = 3.3 \text{ cm}
\]
Now we can calculate the volume of the can:
\[
V_{\text{can}} = \pi (3.3)^2 (19.8)
\]
Calculating \( (3.3)^2 \):
\[
(3.3)^2 = 10.89
\]
Thus:
\[
V_{\text{can}} \approx \pi (10.89)(19.8) \approx 3.14159 \times 215.562 \approx 678.014 \text{ cm}^3
\]
### Step 3: Calculate the space not occupied by the tennis balls
To find the volume of the space not occupied by the tennis balls, we subtract the volume of the balls from the volume of the can:
\[
V_{\text{unused}} = V_{\text{can}} - V_{\text{total balls}}
\]
Substituting the values we calculated:
\[
V_{\text{unused}} = 678.014 - 450.801 \approx 227.213 \text{ cm}^3
\]
### Final Answer
The volume of the space not occupied by the tennis balls is approximately:
\[
\boxed{227.21} \text{ cm}^3
\]