Question
A farmer has a shed that is in the shape of a right circular cylinder, and the interior needs a fresh count of paint (including the floor and ceiling). If the height of the shed is 40 feet, and the diameter is 15 feet, what is the surface area he needs to paint? Use 3.14 for π
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Be sure to show all your work for this question.
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Be sure to show all your work for this question.
Answers
GPT-4o mini
To calculate the total surface area of the farmer's shed, which is a right circular cylinder, we need to find the areas of the lateral surface, the floor, and the ceiling.
### Step 1: Identify the dimensions
- Height \( h = 40 \) feet
- Diameter \( d = 15 \) feet
First, we calculate the radius \( r \):
\[
r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ feet}
\]
### Step 2: Calculate the lateral surface area
The lateral surface area \( A_{\text{lateral}} \) of a cylinder is given by the formula:
\[
A_{\text{lateral}} = 2\pi rh
\]
Substituting the values of \( \pi \), \( r \), and \( h \):
\[
A_{\text{lateral}} = 2 \times 3.14 \times 7.5 \times 40
\]
\[
A_{\text{lateral}} = 6.28 \times 7.5 \times 40
\]
Calculating \( 7.5 \times 40 \):
\[
7.5 \times 40 = 300
\]
Now, calculating \( 6.28 \times 300 \):
\[
A_{\text{lateral}} = 1884 \text{ square feet}
\]
### Step 3: Calculate the area of the top (ceiling) and bottom (floor)
The area \( A_{\text{base}} \) of the base (which is circular) is given by:
\[
A_{\text{base}} = \pi r^2
\]
Calculating this area using the radius:
\[
A_{\text{base}} = 3.14 \times (7.5)^2
\]
Calculating \( (7.5)^2 \):
\[
(7.5)^2 = 56.25
\]
Now,
\[
A_{\text{base}} = 3.14 \times 56.25 = 176.625 \text{ square feet}
\]
Since there are two bases (one for the floor and one for the ceiling), the total area for both bases is:
\[
A_{\text{bases}} = 2 \times A_{\text{base}} = 2 \times 176.625 = 353.25 \text{ square feet}
\]
### Step 4: Calculate the total surface area
Now, we can find the total surface area \( A_{\text{total}} \) that needs to be painted:
\[
A_{\text{total}} = A_{\text{lateral}} + A_{\text{bases}}
\]
Substituting the calculated values:
\[
A_{\text{total}} = 1884 + 353.25 = 2237.25 \text{ square feet}
\]
### Conclusion
The total surface area that the farmer needs to paint is \( 2237.25 \) square feet.
### Step 1: Identify the dimensions
- Height \( h = 40 \) feet
- Diameter \( d = 15 \) feet
First, we calculate the radius \( r \):
\[
r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ feet}
\]
### Step 2: Calculate the lateral surface area
The lateral surface area \( A_{\text{lateral}} \) of a cylinder is given by the formula:
\[
A_{\text{lateral}} = 2\pi rh
\]
Substituting the values of \( \pi \), \( r \), and \( h \):
\[
A_{\text{lateral}} = 2 \times 3.14 \times 7.5 \times 40
\]
\[
A_{\text{lateral}} = 6.28 \times 7.5 \times 40
\]
Calculating \( 7.5 \times 40 \):
\[
7.5 \times 40 = 300
\]
Now, calculating \( 6.28 \times 300 \):
\[
A_{\text{lateral}} = 1884 \text{ square feet}
\]
### Step 3: Calculate the area of the top (ceiling) and bottom (floor)
The area \( A_{\text{base}} \) of the base (which is circular) is given by:
\[
A_{\text{base}} = \pi r^2
\]
Calculating this area using the radius:
\[
A_{\text{base}} = 3.14 \times (7.5)^2
\]
Calculating \( (7.5)^2 \):
\[
(7.5)^2 = 56.25
\]
Now,
\[
A_{\text{base}} = 3.14 \times 56.25 = 176.625 \text{ square feet}
\]
Since there are two bases (one for the floor and one for the ceiling), the total area for both bases is:
\[
A_{\text{bases}} = 2 \times A_{\text{base}} = 2 \times 176.625 = 353.25 \text{ square feet}
\]
### Step 4: Calculate the total surface area
Now, we can find the total surface area \( A_{\text{total}} \) that needs to be painted:
\[
A_{\text{total}} = A_{\text{lateral}} + A_{\text{bases}}
\]
Substituting the calculated values:
\[
A_{\text{total}} = 1884 + 353.25 = 2237.25 \text{ square feet}
\]
### Conclusion
The total surface area that the farmer needs to paint is \( 2237.25 \) square feet.