To calculate the total surface area that the farmer needs to paint, we need to consider both the lateral surface area and the area of the two circular ends (the floor and the ceiling) of the cylindrical shed.
Step 1: Calculate the radius of the cylinder
The diameter of the shed is given as 15 feet. Therefore, the radius (r) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{15 , \text{feet}}{2} = 7.5 , \text{feet} \]
Step 2: Calculate the lateral surface area
The lateral surface area (A_lateral) of a right circular cylinder is given by the formula:
\[ A_{\text{lateral}} = 2 \pi r h \]
Where:
- \(r\) is the radius,
- \(h\) is the height.
Substituting in the known values:
\[ A_{\text{lateral}} = 2 \times 3.14 \times 7.5 \times 40 \]
Calculating step-by-step:
\[ A_{\text{lateral}} = 2 \times 3.14 = 6.28 \]
\[ 6.28 \times 7.5 = 47.1 \]
\[ 47.1 \times 40 = 1884 , \text{square feet} \]
Step 3: Calculate the area of the two circular ends
The area (A_circle) of one circular end is given by the formula:
\[ A_{\text{circle}} = \pi r^2 \]
Therefore, for two circular ends, we have:
\[ A_{\text{ends}} = 2 \times A_{\text{circle}} = 2 \times \pi r^2 \]
Substituting in the known values:
\[ A_{\text{ends}} = 2 \times 3.14 \times (7.5)^2 \]
Calculating \( (7.5)^2 \):
\[ (7.5)^2 = 56.25 \]
Now calculating the area of the two circular ends:
\[ A_{\text{ends}} = 2 \times 3.14 \times 56.25 \]
Calculating step-by-step:
\[ 3.14 \times 56.25 = 176.625 \]
\[ A_{\text{ends}} = 2 \times 176.625 = 353.25 , \text{square feet} \]
Step 4: Calculate the total surface area
Now, we can find the total surface area (A_total) by adding the lateral surface area and the area of the two circular ends:
\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{ends}} \]
Substituting in the values:
\[ A_{\text{total}} = 1884 + 353.25 \]
Calculating the total:
\[ A_{\text{total}} = 2237.25 , \text{square feet} \]
Conclusion
The total surface area that the farmer needs to paint is:
\[ \boxed{2237.25} , \text{square feet} \]