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Answer :
To find the total volume of the grain silo, we need to calculate the volumes of both the cylindrical portion and the hemispherical portion, and then add them together.
### Step 1: Calculate the volume of the cylindrical portion
1. Determine the radius:
- The diameter of the cylinder is given as 4.4 meters.
- Therefore, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
2. Use the formula for the volume of a cylinder:
- The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
- Plug in the given dimensions ([tex]\( \pi \approx 3.14 \)[/tex], [tex]\( r = 2.2 \)[/tex] meters, and [tex]\( h = 6.2 \)[/tex] meters):
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
- Calculate [tex]\( (2.2)^2 = 4.84 \)[/tex]:
[tex]\[
V_{\text{cylinder}} = 3.14 \times 4.84 \times 6.2 = 94.2 \, \text{cubic meters (approximately)}
\][/tex]
### Step 2: Calculate the volume of the hemispherical portion
1. Use the formula for the volume of a hemisphere:
- The formula for the volume of a hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\][/tex]
- Use the same radius ([tex]\( r = 2.2 \)[/tex] meters):
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3
\][/tex]
- Calculate [tex]\( (2.2)^3 = 10.648 \)[/tex]:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 10.648 = 22.3 \, \text{cubic meters (approximately)}
\][/tex]
### Step 3: Calculate the total volume of the silo
- Add the volumes of the cylindrical and hemispherical portions:
[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 is 116.5 cubic meters. The answer is [tex]\( 116.5 \, m^3 \)[/tex].
### Step 1: Calculate the volume of the cylindrical portion
1. Determine the radius:
- The diameter of the cylinder is given as 4.4 meters.
- Therefore, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{4.4}{2} = 2.2 \text{ meters}
\][/tex]
2. Use the formula for the volume of a cylinder:
- The formula for the volume of a cylinder is:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
- Plug in the given dimensions ([tex]\( \pi \approx 3.14 \)[/tex], [tex]\( r = 2.2 \)[/tex] meters, and [tex]\( h = 6.2 \)[/tex] meters):
[tex]\[
V_{\text{cylinder}} = 3.14 \times (2.2)^2 \times 6.2
\][/tex]
- Calculate [tex]\( (2.2)^2 = 4.84 \)[/tex]:
[tex]\[
V_{\text{cylinder}} = 3.14 \times 4.84 \times 6.2 = 94.2 \, \text{cubic meters (approximately)}
\][/tex]
### Step 2: Calculate the volume of the hemispherical portion
1. Use the formula for the volume of a hemisphere:
- The formula for the volume of a hemisphere is:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\][/tex]
- Use the same radius ([tex]\( r = 2.2 \)[/tex] meters):
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3
\][/tex]
- Calculate [tex]\( (2.2)^3 = 10.648 \)[/tex]:
[tex]\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 10.648 = 22.3 \, \text{cubic meters (approximately)}
\][/tex]
### Step 3: Calculate the total volume of the silo
- Add the volumes of the cylindrical and hemispherical portions:
[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 is 116.5 cubic meters. The answer is [tex]\( 116.5 \, m^3 \)[/tex].
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