To calculate the surface area of a right circular cylinder (including the floor and ceiling), we need to consider the lateral surface area as well as the areas of the top and bottom circles (bases).
Step 1: Find the radius
The diameter of the shed is given as 15 feet. The radius (r) is half of the diameter:
\[ r = \frac{\text{diameter}}{2} = \frac{15}{2} = 7.5 \text{ feet} \]
Step 2: Find the height
The height (h) of the shed is given as 40 feet.
Step 3: Calculate the lateral surface area
The formula for the lateral surface area (A_lateral) of a cylinder is:
\[ A_{\text{lateral}} = 2\pi r h \]
Substituting in the values we have:
\[ A_{\text{lateral}} = 2 \times 3.14 \times 7.5 \times 40 \]
Calculating step-by-step:
- Calculate \(2 \times 3.14 = 6.28\).
- Calculate \(6.28 \times 7.5 = 47.1\).
- Calculate \(47.1 \times 40 = 1884\).
Thus,
\[ A_{\text{lateral}} = 1884 \text{ square feet} \]
Step 4: Calculate the area of the top and bottom circles
The area (A) of one circle is given by the formula:
\[ A = \pi r^2 \]
Now, substituting the radius we found:
\[ A = 3.14 \times (7.5)^2 \]
Calculating step-by-step:
- Calculate \((7.5)^2 = 56.25\).
- Calculate \(3.14 \times 56.25 = 176.625\).
Thus,
\[ A = 176.625 \text{ square feet} \]
Since the cylinder has two circular bases (one on the top and one on the bottom), the total area for both bases is:
\[ A_{\text{bases}} = 2 \times 176.625 = 353.25 \text{ square feet} \]
Step 5: Total surface area
Now we can find the total surface area (A_total) by adding the lateral surface area and the area of the bases:
\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{bases}} \]
Substituting in the values we calculated:
\[ A_{\text{total}} = 1884 + 353.25 = 2237.25 \text{ square feet} \]
Final Answer
The total surface area that needs to be painted is
\[ \boxed{2237.25} \text{ square feet} \]