Answer :
* Calculates the radius of the silo using the given diameter: $r = 4.4 / 2 = 2.2$ m.
* Computes the volume of the cylindrical portion: $V_{cylinder} = \pi r^2 h = 3.14 \times (2.2)^2 \times 6.2 \approx 94.2 m^3$.
* Calculates the volume of the hemispherical portion: $V_{hemisphere} = (2/3) \pi r^3 = (2/3) \times 3.14 \times (2.2)^3 \approx 22.3 m^3$.
* Determines the total volume by summing the two volumes and rounding: $V_{total} = 94.2 + 22.3 = 116.5 m^3$. The final answer is $\boxed{116.5 m^3}$.
### Explanation
1. Problem Analysis and Given Data
The grain silo consists of a cylinder and a hemisphere. We need to find the total volume of the silo, which is the sum of the volumes of the cylinder and the hemisphere. The diameter of the silo is 4.4 meters, so the radius is half of that, which is 2.2 meters. The height of the cylindrical part is 6.2 meters. We'll use 3.14 as the value for $\pi$.
2. Volume of the Cylinder
First, let's calculate the volume of the cylindrical portion of the silo. The formula for the volume of a cylinder is $V_{cylinder} = \pi r^2 h$, where $r$ is the radius and $h$ is the height. In this case, $r = 2.2$ meters and $h = 6.2$ meters. So, $V_{cylinder} = 3.14 \times (2.2)^2 \times 6.2$.
3. Volume of the Hemisphere
Now, let's calculate the volume of the hemisphere. The formula for the volume of a sphere is $V_{sphere} = \frac{4}{3} \pi r^3$, so the volume of a hemisphere is half of that, which is $V_{hemisphere} = \frac{2}{3} \pi r^3$. In this case, $r = 2.2$ meters. So, $V_{hemisphere} = \frac{2}{3} \times 3.14 \times (2.2)^3$.
4. Calculating Volumes
Now we calculate the values:
$V_{cylinder} = 3.14 \times (2.2)^2 \times 6.2 = 3.14 \times 4.84 \times 6.2 = 15.20 \times 6.2 = 94.244 \approx 94.2 m^3$
$V_{hemisphere} = \frac{2}{3} \times 3.14 \times (2.2)^3 = \frac{2}{3} \times 3.14 \times 10.648 = \frac{2}{3} \times 33.43 \approx 22.3 m^3$
5. Total Volume Calculation
Now, let's add the volumes of the cylinder and the hemisphere to find the total volume of the silo:
$V_{total} = V_{cylinder} + V_{hemisphere} = 94.244 + 22.286 \approx 116.53 m^3$. Rounding to the nearest tenth of a cubic meter, we get $116.5 m^3$.
6. Final Answer
The approximate total volume of the silo is $116.5 m^3$.
### Examples
Understanding how to calculate the volume of silos is crucial in agriculture for storing grains efficiently. Farmers and agricultural engineers use these calculations to determine the capacity needed for grain storage, optimize space, and manage resources effectively. This ensures proper storage and preservation of grains, which is vital for food supply and economic stability.
* Computes the volume of the cylindrical portion: $V_{cylinder} = \pi r^2 h = 3.14 \times (2.2)^2 \times 6.2 \approx 94.2 m^3$.
* Calculates the volume of the hemispherical portion: $V_{hemisphere} = (2/3) \pi r^3 = (2/3) \times 3.14 \times (2.2)^3 \approx 22.3 m^3$.
* Determines the total volume by summing the two volumes and rounding: $V_{total} = 94.2 + 22.3 = 116.5 m^3$. The final answer is $\boxed{116.5 m^3}$.
### Explanation
1. Problem Analysis and Given Data
The grain silo consists of a cylinder and a hemisphere. We need to find the total volume of the silo, which is the sum of the volumes of the cylinder and the hemisphere. The diameter of the silo is 4.4 meters, so the radius is half of that, which is 2.2 meters. The height of the cylindrical part is 6.2 meters. We'll use 3.14 as the value for $\pi$.
2. Volume of the Cylinder
First, let's calculate the volume of the cylindrical portion of the silo. The formula for the volume of a cylinder is $V_{cylinder} = \pi r^2 h$, where $r$ is the radius and $h$ is the height. In this case, $r = 2.2$ meters and $h = 6.2$ meters. So, $V_{cylinder} = 3.14 \times (2.2)^2 \times 6.2$.
3. Volume of the Hemisphere
Now, let's calculate the volume of the hemisphere. The formula for the volume of a sphere is $V_{sphere} = \frac{4}{3} \pi r^3$, so the volume of a hemisphere is half of that, which is $V_{hemisphere} = \frac{2}{3} \pi r^3$. In this case, $r = 2.2$ meters. So, $V_{hemisphere} = \frac{2}{3} \times 3.14 \times (2.2)^3$.
4. Calculating Volumes
Now we calculate the values:
$V_{cylinder} = 3.14 \times (2.2)^2 \times 6.2 = 3.14 \times 4.84 \times 6.2 = 15.20 \times 6.2 = 94.244 \approx 94.2 m^3$
$V_{hemisphere} = \frac{2}{3} \times 3.14 \times (2.2)^3 = \frac{2}{3} \times 3.14 \times 10.648 = \frac{2}{3} \times 33.43 \approx 22.3 m^3$
5. Total Volume Calculation
Now, let's add the volumes of the cylinder and the hemisphere to find the total volume of the silo:
$V_{total} = V_{cylinder} + V_{hemisphere} = 94.244 + 22.286 \approx 116.53 m^3$. Rounding to the nearest tenth of a cubic meter, we get $116.5 m^3$.
6. Final Answer
The approximate total volume of the silo is $116.5 m^3$.
### Examples
Understanding how to calculate the volume of silos is crucial in agriculture for storing grains efficiently. Farmers and agricultural engineers use these calculations to determine the capacity needed for grain storage, optimize space, and manage resources effectively. This ensures proper storage and preservation of grains, which is vital for food supply and economic stability.
