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Sagot :
To determine how much of the canister's volume is unoccupied by the tennis balls, follow these detailed steps:
1. Calculate the volume of the canister:
- The canister is a cylinder, so you use the formula for the volume of a cylinder, [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- Given:
- Radius [tex]\( r = 1.5 \)[/tex] inches
- Height [tex]\( h = 7 \)[/tex] inches
- [tex]\(\pi = 3.14\)[/tex]
- Plug in the values:
[tex]\[ V_{\text{canister}} = 3.14 \times (1.5)^2 \times 7 \][/tex]
- Calculate [tex]\((1.5)^2\)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
- Then:
[tex]\[ V_{\text{canister}} = 3.14 \times 2.25 \times 7 = 49.4625 \approx 49.46 \text{ cubic inches} \][/tex]
2. Calculate the volume of one tennis ball:
- The tennis ball is a sphere, so you use the formula for the volume of a sphere, [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- The diameter of one tennis ball is 2.25 inches, so the radius [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{2.25}{2} = 1.125 \text{ inches} \][/tex]
- Plug in the values:
[tex]\[ V_{\text{ball}} = \frac{4}{3} \times 3.14 \times (1.125)^3 \][/tex]
- Calculate [tex]\( (1.125)^3 \)[/tex]:
[tex]\[ (1.125)^3 \approx 1.423828125 \][/tex]
- Then:
[tex]\[ V_{\text{ball}} = \frac{4}{3} \times 3.14 \times 1.423828125 \approx 5.958 \text{ cubic inches} \][/tex]
- Round to two decimal places:
[tex]\[ V_{\text{ball}} \approx 5.96 \text{ cubic inches} \][/tex]
3. Calculate the total volume occupied by three tennis balls:
- Since there are 3 tennis balls:
[tex]\[ V_{\text{3 balls}} = 3 \times V_{\text{ball}} = 3 \times 5.96 \approx 17.88 \text{ cubic inches} \][/tex]
4. Calculate the unoccupied volume of the canister:
- The unoccupied volume is the volume of the canister minus the total volume occupied by the three tennis balls:
[tex]\[ V_{\text{unoccupied}} = V_{\text{canister}} - V_{\text{3 balls}} \][/tex]
- Given:
[tex]\[ V_{\text{canister}} \approx 49.46 \text{ cubic inches} \][/tex]
[tex]\[ V_{\text{3 balls}} \approx 17.88 \text{ cubic inches} \][/tex]
- Subtract to find the unoccupied volume:
[tex]\[ V_{\text{unoccupied}} = 49.46 - 17.88 \approx 31.57 \text{ cubic inches} \][/tex]
So, the volume of the canister that is unoccupied by the tennis balls is approximately 31.57 cubic inches.
The correct answer among the provided options is:
31.57 in[tex]\(^3\)[/tex].
1. Calculate the volume of the canister:
- The canister is a cylinder, so you use the formula for the volume of a cylinder, [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- Given:
- Radius [tex]\( r = 1.5 \)[/tex] inches
- Height [tex]\( h = 7 \)[/tex] inches
- [tex]\(\pi = 3.14\)[/tex]
- Plug in the values:
[tex]\[ V_{\text{canister}} = 3.14 \times (1.5)^2 \times 7 \][/tex]
- Calculate [tex]\((1.5)^2\)[/tex]:
[tex]\[ (1.5)^2 = 2.25 \][/tex]
- Then:
[tex]\[ V_{\text{canister}} = 3.14 \times 2.25 \times 7 = 49.4625 \approx 49.46 \text{ cubic inches} \][/tex]
2. Calculate the volume of one tennis ball:
- The tennis ball is a sphere, so you use the formula for the volume of a sphere, [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- The diameter of one tennis ball is 2.25 inches, so the radius [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{2.25}{2} = 1.125 \text{ inches} \][/tex]
- Plug in the values:
[tex]\[ V_{\text{ball}} = \frac{4}{3} \times 3.14 \times (1.125)^3 \][/tex]
- Calculate [tex]\( (1.125)^3 \)[/tex]:
[tex]\[ (1.125)^3 \approx 1.423828125 \][/tex]
- Then:
[tex]\[ V_{\text{ball}} = \frac{4}{3} \times 3.14 \times 1.423828125 \approx 5.958 \text{ cubic inches} \][/tex]
- Round to two decimal places:
[tex]\[ V_{\text{ball}} \approx 5.96 \text{ cubic inches} \][/tex]
3. Calculate the total volume occupied by three tennis balls:
- Since there are 3 tennis balls:
[tex]\[ V_{\text{3 balls}} = 3 \times V_{\text{ball}} = 3 \times 5.96 \approx 17.88 \text{ cubic inches} \][/tex]
4. Calculate the unoccupied volume of the canister:
- The unoccupied volume is the volume of the canister minus the total volume occupied by the three tennis balls:
[tex]\[ V_{\text{unoccupied}} = V_{\text{canister}} - V_{\text{3 balls}} \][/tex]
- Given:
[tex]\[ V_{\text{canister}} \approx 49.46 \text{ cubic inches} \][/tex]
[tex]\[ V_{\text{3 balls}} \approx 17.88 \text{ cubic inches} \][/tex]
- Subtract to find the unoccupied volume:
[tex]\[ V_{\text{unoccupied}} = 49.46 - 17.88 \approx 31.57 \text{ cubic inches} \][/tex]
So, the volume of the canister that is unoccupied by the tennis balls is approximately 31.57 cubic inches.
The correct answer among the provided options is:
31.57 in[tex]\(^3\)[/tex].
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