To solve this problem, we'll break it down into manageable steps:
1. Understand the Shape and Dimensions:
- The silo has a cylindrical body topped with a hemisphere.
- The diameter of the silo is 3.8 meters.
- The total height of the silo, including the hemisphere, is 9.3 meters.
2. Calculate the Radius:
- The radius of both the cylinder and the hemisphere is half the diameter.
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{3.8}{2} = 1.9 \text{ meters}
\][/tex]
3. Calculate the Height of the Cylindrical Part:
- The height of the cylindrical part is the total height minus the radius of the hemisphere (since the hemisphere’s height is equal to its radius).
[tex]\[
\text{cylinder height} = \text{total height} - \text{radius} = 9.3 - 1.9 = 7.4 \text{ meters}
\][/tex]
4. Surface Area of the Cylindrical Part:
- The surface area (excluding the base) of the cylindrical part is calculated using the formula [tex]\(2 \pi r h\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
[tex]\[
\text{cylindrical surface area} = 2 \pi \times 1.9 \times 7.4 = 88.34158541894499 \text{ square meters}
\][/tex]
5. Surface Area of the Hemisphere:
- The surface area of a hemisphere is [tex]\(2 \pi r^2\)[/tex].
[tex]\[
\text{hemisphere surface area} = 2 \pi \times (1.9)^2 = 22.682298958918306 \text{ square meters}
\][/tex]
6. Total Surface Area:
- Add the surface areas of the cylindrical part and the hemisphere to get the total surface area that needs to be painted.
[tex]\[
\text{total surface area} = 88.34158541894499 + 22.682298958918306 = 111.0238843778633 \text{ square meters}
\][/tex]
Hence, the number of square meters that need to be painted on the silo is approximately 111.024 square meters.
