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Answer :
To find the total volume of the grain silo, which consists of a cylindrical portion and a hemispherical portion, we'll follow these steps:
1. Calculate the Volume of the Cylindrical Portion:
- The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi \times r^2 \times h
\][/tex]
- The diameter of the silo is given as 4.4 meters, so the radius [tex]\( r \)[/tex] is half of that:
[tex]\[
r = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
- The height [tex]\( h \)[/tex] of the cylindrical portion is 6.2 meters.
- Using [tex]\(\pi = 3.14\)[/tex], the volume of the cylinder is:
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
- After calculating, the cylindrical volume is approximately 94.2 cubic meters.
2. Calculate the Volume of the Hemispherical Portion:
- The formula for the volume of a hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times \pi \times r^3
\][/tex]
- We have the same radius [tex]\( r = 2.2 \)[/tex] meters.
- Using [tex]\(\pi = 3.14\)[/tex], the volume of the hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3
\][/tex]
- After calculating, the hemispherical volume is approximately 22.3 cubic meters.
3. Calculate the Total Volume of the Silo:
- Add the volume of the cylindrical portion and the hemispherical portion:
[tex]\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}}
\][/tex]
- Which is approximately:
[tex]\[
V_{\text{total}} = 94.2 + 22.3 = 116.5 \text{ cubic meters}
\][/tex]
4. Round the Total Volume:
- The total volume is already rounded to the nearest tenth, which is 116.5 cubic meters.
The approximate total volume of the silo is [tex]\(116.5\)[/tex] cubic meters. Thus, the correct option is [tex]\( \boxed{116.5 \, m^3} \)[/tex].
1. Calculate the Volume of the Cylindrical Portion:
- The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi \times r^2 \times h
\][/tex]
- The diameter of the silo is given as 4.4 meters, so the radius [tex]\( r \)[/tex] is half of that:
[tex]\[
r = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
- The height [tex]\( h \)[/tex] of the cylindrical portion is 6.2 meters.
- Using [tex]\(\pi = 3.14\)[/tex], the volume of the cylinder is:
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
- After calculating, the cylindrical volume is approximately 94.2 cubic meters.
2. Calculate the Volume of the Hemispherical Portion:
- The formula for the volume of a hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times \pi \times r^3
\][/tex]
- We have the same radius [tex]\( r = 2.2 \)[/tex] meters.
- Using [tex]\(\pi = 3.14\)[/tex], the volume of the hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3
\][/tex]
- After calculating, the hemispherical volume is approximately 22.3 cubic meters.
3. Calculate the Total Volume of the Silo:
- Add the volume of the cylindrical portion and the hemispherical portion:
[tex]\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}}
\][/tex]
- Which is approximately:
[tex]\[
V_{\text{total}} = 94.2 + 22.3 = 116.5 \text{ cubic meters}
\][/tex]
4. Round the Total Volume:
- The total volume is already rounded to the nearest tenth, which is 116.5 cubic meters.
The approximate total volume of the silo is [tex]\(116.5\)[/tex] cubic meters. Thus, the correct option is [tex]\( \boxed{116.5 \, m^3} \)[/tex].
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