To determine the minimum number of gallons needed to paint the exterior surface of the farm silo, we need to follow several steps. We must calculate the surface area of the silo that needs painting (excluding the bottom) and then use the paint coverage rate to find how much paint is required.
First, we analyze the dimensions of the silo:
- The radius of the silo base is [tex]\(10\)[/tex] feet.
- The height of the silo is [tex]\(50\)[/tex] feet.
- We use [tex]\(\pi = 3.14\)[/tex].
The surface area to be painted consists of two parts:
1. The lateral surface area of the cylindrical part.
2. The area of the circular top.
### Step 1: Calculate the lateral surface area of the cylindrical part
The lateral surface area [tex]\(A_{\text{cylindrical}}\)[/tex] of a cylinder is given by:
[tex]\[ A_{\text{cylindrical}} = 2 \pi r h \][/tex]
Substituting [tex]\(r = 10\)[/tex] feet and [tex]\(h = 50\)[/tex] feet:
[tex]\[ A_{\text{cylindrical}} = 2 \times 3.14 \times 10 \times 50 \][/tex]
[tex]\[ A_{\text{cylindrical}} = 3140 \, \text{square feet} \][/tex]
### Step 2: Calculate the area of the circular top
The area [tex]\(A_{\text{top}}\)[/tex] of the circular top is given by:
[tex]\[ A_{\text{top}} = \pi r^2 \][/tex]
Substituting [tex]\(r = 10\)[/tex] feet:
[tex]\[ A_{\text{top}} = 3.14 \times 10^2 \][/tex]
[tex]\[ A_{\text{top}} = 3.14 \times 100 \][/tex]
[tex]\[ A_{\text{top}} = 314 \, \text{square feet} \][/tex]
### Step 3: Calculate the total surface area to be painted
The total surface area [tex]\(A_{\text{total}}\)[/tex] to be painted is the sum of the lateral surface area and the top area:
[tex]\[ A_{\text{total}} = A_{\text{cylindrical}} + A_{\text{top}} \][/tex]
[tex]\[ A_{\text{total}} = 3140 + 314 \][/tex]
[tex]\[ A_{\text{total}} = 3454 \, \text{square feet} \][/tex]
### Step 4: Calculate the number of gallons needed
One gallon of paint covers [tex]\(224\)[/tex] square feet. To find the total number of gallons needed:
[tex]\[ \text{Gallons needed} = \frac{A_{\text{total}}}{\text{paint coverage per gallon}} \][/tex]
[tex]\[ \text{Gallons needed} = \frac{3454}{224} \][/tex]
[tex]\[ \text{Gallons needed} \approx 15.42 \][/tex]
Since only whole gallons can be purchased, we need to round up to the nearest whole number:
[tex]\[ \text{Minimum number of gallons needed} = 16 \][/tex]
### Conclusion
The minimum number of gallons needed to paint the silo, keeping in mind that the bottom is not painted, is:
[tex]\[ \boxed{16} \][/tex]
