Answer :
We are given that the grain silo consists of a cylinder and a hemisphere. The key dimensions are:
- The diameter of both the cylinder and the hemisphere is 4.4 meters, so the radius is
[tex]$$r = \frac{4.4}{2} = 2.2 \text{ meters}.$$[/tex]
- The height of the cylindrical portion is 6.2 meters.
- We use the value [tex]$\pi = 3.14$[/tex].
Step 1. Calculate the Volume of the Cylindrical Portion
The volume of a cylinder is given by
[tex]$$
V_{\text{cyl}} = \pi r^2 h.
$$[/tex]
Substitute [tex]$r = 2.2\text{ m}$[/tex] and [tex]$h = 6.2\text{ m}$[/tex]:
[tex]$$
V_{\text{cyl}} = 3.14 \times (2.2)^2 \times 6.2.
$$[/tex]
Evaluating [tex]$(2.2)^2$[/tex]:
[tex]$$
(2.2)^2 = 4.84.
$$[/tex]
Thus,
[tex]$$
V_{\text{cyl}} = 3.14 \times 4.84 \times 6.2 \approx 94.22512 \text{ m}^3.
$$[/tex]
Step 2. Calculate the Volume of the Hemispherical Portion
The volume of a full sphere is
[tex]$$
V_{\text{sphere}} = \frac{4}{3}\pi r^3,
$$[/tex]
so the volume of a hemisphere (half of a sphere) is
[tex]$$
V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{4}{3}\pi r^3\right) = \frac{2}{3} \pi r^3.
$$[/tex]
Substitute [tex]$r = 2.2\text{ m}$[/tex]:
[tex]$$
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3.
$$[/tex]
Evaluating [tex]$(2.2)^3$[/tex]:
[tex]$$
(2.2)^3 \approx 10.648.
$$[/tex]
Thus,
[tex]$$
V_{\text{hemisphere}} \approx \frac{2}{3} \times 3.14 \times 10.648 \approx 22.28981333 \text{ m}^3.
$$[/tex]
Step 3. Calculate the Total Volume of the Silo
Add the volumes of the cylindrical and hemispherical portions:
[tex]$$
V_{\text{total}} = V_{\text{cyl}} + V_{\text{hemisphere}} \approx 94.22512 + 22.28981 \approx 116.51493 \text{ m}^3.
$$[/tex]
Rounding to the nearest tenth,
[tex]$$
V_{\text{total}} \approx 116.5 \text{ m}^3.
$$[/tex]
Final Answer: The approximate total volume of the silo is [tex]$$116.5 \text{ m}^3.$$[/tex]
- The diameter of both the cylinder and the hemisphere is 4.4 meters, so the radius is
[tex]$$r = \frac{4.4}{2} = 2.2 \text{ meters}.$$[/tex]
- The height of the cylindrical portion is 6.2 meters.
- We use the value [tex]$\pi = 3.14$[/tex].
Step 1. Calculate the Volume of the Cylindrical Portion
The volume of a cylinder is given by
[tex]$$
V_{\text{cyl}} = \pi r^2 h.
$$[/tex]
Substitute [tex]$r = 2.2\text{ m}$[/tex] and [tex]$h = 6.2\text{ m}$[/tex]:
[tex]$$
V_{\text{cyl}} = 3.14 \times (2.2)^2 \times 6.2.
$$[/tex]
Evaluating [tex]$(2.2)^2$[/tex]:
[tex]$$
(2.2)^2 = 4.84.
$$[/tex]
Thus,
[tex]$$
V_{\text{cyl}} = 3.14 \times 4.84 \times 6.2 \approx 94.22512 \text{ m}^3.
$$[/tex]
Step 2. Calculate the Volume of the Hemispherical Portion
The volume of a full sphere is
[tex]$$
V_{\text{sphere}} = \frac{4}{3}\pi r^3,
$$[/tex]
so the volume of a hemisphere (half of a sphere) is
[tex]$$
V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{4}{3}\pi r^3\right) = \frac{2}{3} \pi r^3.
$$[/tex]
Substitute [tex]$r = 2.2\text{ m}$[/tex]:
[tex]$$
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times (2.2)^3.
$$[/tex]
Evaluating [tex]$(2.2)^3$[/tex]:
[tex]$$
(2.2)^3 \approx 10.648.
$$[/tex]
Thus,
[tex]$$
V_{\text{hemisphere}} \approx \frac{2}{3} \times 3.14 \times 10.648 \approx 22.28981333 \text{ m}^3.
$$[/tex]
Step 3. Calculate the Total Volume of the Silo
Add the volumes of the cylindrical and hemispherical portions:
[tex]$$
V_{\text{total}} = V_{\text{cyl}} + V_{\text{hemisphere}} \approx 94.22512 + 22.28981 \approx 116.51493 \text{ m}^3.
$$[/tex]
Rounding to the nearest tenth,
[tex]$$
V_{\text{total}} \approx 116.5 \text{ m}^3.
$$[/tex]
Final Answer: The approximate total volume of the silo is [tex]$$116.5 \text{ m}^3.$$[/tex]
