Question
23 in
40 in height
Find the surface area of this cylinder.
6,283 in 2
1,210 in 2
16,283 in 2
10.210 in 2
40 in height
Find the surface area of this cylinder.
6,283 in 2
1,210 in 2
16,283 in 2
10.210 in 2
Answers
GPT-4o mini
To calculate the surface area of a cylinder, we use the formula:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder,
- \( \pi \) is approximately 3.14159.
Given:
- Diameter = 23 in, so the radius \( r = \frac{23}{2} = 11.5 \) in
- Height \( h = 40 \) in
Now plug in the values:
1. Calculate \( h + r \):
\[
h + r = 40 + 11.5 = 51.5 \text{ in}
\]
2. Find the surface area:
\[
\text{Surface Area} = 2\pi (11.5)(51.5)
\]
Now calculate \( 2\pi \):
\[
2\pi \approx 6.28318
\]
Now multiply:
\[
\text{Surface Area} \approx 6.28318 \times 11.5 \times 51.5
\]
Calculating that:
\[
11.5 \times 51.5 \approx 593.25
\]
and then:
\[
6.28318 \times 593.25 \approx 3,724.46
\]
Thus, the surface area:
\[
\text{Surface Area} \approx 3,724.46 \text{ in}^2
\]
It appears I made an error in not computing the full surface area properly; let’s do it properly according to the complete formula:
1. Calculate the lateral surface area, which is \( 2\pi rh \):
\[
2\pi rh = 2\pi (11.5)(40) = 6.28318 \times 11.5 \times 40 \approx 6.28318 \times 460 \approx 2,887.45 \text{ in}^2
\]
2. Calculate the area of the two bases (top and bottom):
\[
2\pi r^2 = 2\pi (11.5)^2
\]
Calculate \( (11.5)^2 \):
\[
11.5^2 = 132.25
\]
Then:
\[
2\pi (132.25) \approx 6.28318 \times 132.25 \approx 832.94 \text{ in}^2
\]
3. Adding both surface areas together:
\[
\text{Total Surface Area} \approx 2,887.45 + 832.94 \approx 3,720.39 \text{ in}^2
\]
It seems none of the options match this. The most likely candidate based on incorrect calculations may suggest revisiting the context or seeing through other approximations. Unfortunately, based on basic calculations, the value does not correlate with given choices.
Please check potential rounding methods or further details, as numerical resolution might also derive different spheres or approximations. The calculations suggest that a review could correct any listed surface areas.
\[
\text{Surface Area} = 2\pi r(h + r)
\]
where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder,
- \( \pi \) is approximately 3.14159.
Given:
- Diameter = 23 in, so the radius \( r = \frac{23}{2} = 11.5 \) in
- Height \( h = 40 \) in
Now plug in the values:
1. Calculate \( h + r \):
\[
h + r = 40 + 11.5 = 51.5 \text{ in}
\]
2. Find the surface area:
\[
\text{Surface Area} = 2\pi (11.5)(51.5)
\]
Now calculate \( 2\pi \):
\[
2\pi \approx 6.28318
\]
Now multiply:
\[
\text{Surface Area} \approx 6.28318 \times 11.5 \times 51.5
\]
Calculating that:
\[
11.5 \times 51.5 \approx 593.25
\]
and then:
\[
6.28318 \times 593.25 \approx 3,724.46
\]
Thus, the surface area:
\[
\text{Surface Area} \approx 3,724.46 \text{ in}^2
\]
It appears I made an error in not computing the full surface area properly; let’s do it properly according to the complete formula:
1. Calculate the lateral surface area, which is \( 2\pi rh \):
\[
2\pi rh = 2\pi (11.5)(40) = 6.28318 \times 11.5 \times 40 \approx 6.28318 \times 460 \approx 2,887.45 \text{ in}^2
\]
2. Calculate the area of the two bases (top and bottom):
\[
2\pi r^2 = 2\pi (11.5)^2
\]
Calculate \( (11.5)^2 \):
\[
11.5^2 = 132.25
\]
Then:
\[
2\pi (132.25) \approx 6.28318 \times 132.25 \approx 832.94 \text{ in}^2
\]
3. Adding both surface areas together:
\[
\text{Total Surface Area} \approx 2,887.45 + 832.94 \approx 3,720.39 \text{ in}^2
\]
It seems none of the options match this. The most likely candidate based on incorrect calculations may suggest revisiting the context or seeing through other approximations. Unfortunately, based on basic calculations, the value does not correlate with given choices.
Please check potential rounding methods or further details, as numerical resolution might also derive different spheres or approximations. The calculations suggest that a review could correct any listed surface areas.