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Answer :
To find the total volume of the grain silo, which is composed of a cylinder topped with a hemisphere, we'll follow these steps:
1. Determine the radius:
The grain silo has a diameter of 4.4 meters. Radius is half of the diameter:
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
2. Calculate the volume of the cylindrical portion:
The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi \times \text{radius}^2 \times \text{height}
\][/tex]
Given the height of the cylindrical portion is 6.2 meters and [tex]\(\pi \approx 3.14\)[/tex], we have:
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
[tex]\[
V_{\text{cylinder}} = 3.14 \times 4.84 \times 6.2 = 94.2 \text{ cubic meters}
\][/tex]
3. Calculate the volume of the hemisphere:
The formula for the volume of a sphere is:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi \times \text{radius}^3
\][/tex]
Since we only have a hemisphere (half of a sphere), we divide by 2:
[tex]\[
V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \times \pi \times \text{radius}^3 = \frac{2}{3} \times \pi \times (2.2)^3
\][/tex]
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 10.648 = 22.3 \text{ cubic meters}
\][/tex]
4. Add the volumes of the cylindrical portion and the hemisphere to find the total volume of the silo:
[tex]\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 94.2 + 22.3 = 116.5 \text{ cubic meters}
\][/tex]
Therefore, the approximate total volume of the silo, rounded to the nearest tenth of a cubic meter, is 116.5 cubic meters. This matches with the choice 116.5 m³.
1. Determine the radius:
The grain silo has a diameter of 4.4 meters. Radius is half of the diameter:
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
2. Calculate the volume of the cylindrical portion:
The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi \times \text{radius}^2 \times \text{height}
\][/tex]
Given the height of the cylindrical portion is 6.2 meters and [tex]\(\pi \approx 3.14\)[/tex], we have:
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
[tex]\[
V_{\text{cylinder}} = 3.14 \times 4.84 \times 6.2 = 94.2 \text{ cubic meters}
\][/tex]
3. Calculate the volume of the hemisphere:
The formula for the volume of a sphere is:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi \times \text{radius}^3
\][/tex]
Since we only have a hemisphere (half of a sphere), we divide by 2:
[tex]\[
V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \times \pi \times \text{radius}^3 = \frac{2}{3} \times \pi \times (2.2)^3
\][/tex]
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 10.648 = 22.3 \text{ cubic meters}
\][/tex]
4. Add the volumes of the cylindrical portion and the hemisphere to find the total volume of the silo:
[tex]\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 94.2 + 22.3 = 116.5 \text{ cubic meters}
\][/tex]
Therefore, the approximate total volume of the silo, rounded to the nearest tenth of a cubic meter, is 116.5 cubic meters. This matches with the choice 116.5 m³.
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