To find the total surface area that the farmer needs to paint, we need to calculate the surface area of both the lateral area of the circular cylinder and the areas of the top and bottom circular bases.
Step 1: Calculate the radius of the cylinder.
The diameter of the cylinder is given as 15 feet, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{15 \text{ feet}}{2} = 7.5 \text{ feet} \]
Step 2: Calculate the lateral surface area.
The formula for the lateral surface area \( A_L \) of a cylinder is given by:
\[ A_L = 2\pi rh \]
where \( h \) is the height of the cylinder.
Using \( r = 7.5 \) feet and \( h = 40 \) feet, and substituting \( \pi \) with 3.14:
\[ A_L = 2 \times 3.14 \times 7.5 \times 40 \]
Calculating step by step:
- Calculate \( 2 \times 3.14 = 6.28 \)
- Calculate \( 6.28 \times 7.5 = 47.1 \)
- Finally, calculate \( 47.1 \times 40 = 1884 \) square feet.
So, the lateral surface area is:
\[ A_L = 1884 \text{ square feet} \]
Step 3: Calculate the area of the top and bottom circular bases.
The area \( A_B \) of one circular base is given by:
\[ A_B = \pi r^2 \]
Calculating the area of one base:
\[ A_B = 3.14 \times (7.5)^2 \]
Calculating \( (7.5)^2 = 56.25 \):
\[ A_B = 3.14 \times 56.25 = 176.625 \text{ square feet} \]
Since there are two bases (top and bottom), the total area of the bases \( A_{bases} \) is:
\[ A_{bases} = 2 \times A_B = 2 \times 176.625 = 353.25 \text{ square feet} \]
Step 4: Calculate the total surface area to be painted.
The total surface area \( A_{total} \) is the sum of the lateral area and the area of the bases:
\[ A_{total} = A_L + A_{bases} \] \[ A_{total} = 1884 + 353.25 = 2237.25 \text{ square feet} \]
Final Answer:
The total surface area the farmer needs to paint is:
\[ \boxed{2237.25} \text{ square feet} \]