Question
Find the surface area of the cylinder.
Radius: 3 yd
Height: 4 yd
A. 45.7 yds^2
B. 51.84 yds^2
C. 37.8 yds^2
D. 14.14 yds^2
Radius: 3 yd
Height: 4 yd
A. 45.7 yds^2
B. 51.84 yds^2
C. 37.8 yds^2
D. 14.14 yds^2
Answers
GPT-4o mini
To find the surface area of a cylinder, we can use the formula:
\[
SA = 2\pi r(h + r)
\]
where \( SA \) is the surface area, \( r \) is the radius, and \( h \) is the height.
Given:
- Radius \( r = 3 \) yd
- Height \( h = 4 \) yd
Now we can substitute the values into the formula:
\[
SA = 2\pi(3)(4 + 3)
\]
Calculating inside the parentheses first:
\[
h + r = 4 + 3 = 7
\]
Now plug that back into the formula:
\[
SA = 2\pi(3)(7)
\]
\[
SA = 2\pi(21)
\]
\[
SA = 42\pi
\]
Now we can use an approximate value for \( \pi \) (approximately 3.14) to calculate the numerical value:
\[
SA \approx 42 \times 3.14 = 131.88 \text{ yds}^2
\]
This is not in your options. Let's calculate it in a detailed way:
1. The lateral surface area of the cylinder is given by:
\[
LSA = 2\pi rh = 2\pi(3)(4) = 24\pi
\]
which is approximately \( 24 \times 3.14 \approx 75.36 \text{ yds}^2 \).
2. The area of the top and bottom circles is:
\[
A_{circles} = 2\pi r^2 = 2\pi(3^2) = 2\pi(9) = 18\pi
\]
which is approximately \( 18 \times 3.14 \approx 56.52 \text{ yds}^2 \).
3. Now, add both areas together to get the total surface area:
\[
SA = LSA + A_{circles} = 24\pi + 18\pi = 42\pi \approx 131.88 \text{ yds}^2
\]
Since we have calculated both lateral and circular bases separately and reached the total area which is not there in the provided options, I will summarize. The provided multiple choice answers do not include the correct calculated surface area for the cylinder of radius 3 yards and height 4 yards.
Assuming a different interpretation or error since it appears that none of your choices directly match the surface area calculated, it’s crucial we verify if the question or specifications are correct or reach the right approximations. If truly these were the only options presented, please reconfirm the numbers, or look for potential discrepancies in the specifications.
\[
SA = 2\pi r(h + r)
\]
where \( SA \) is the surface area, \( r \) is the radius, and \( h \) is the height.
Given:
- Radius \( r = 3 \) yd
- Height \( h = 4 \) yd
Now we can substitute the values into the formula:
\[
SA = 2\pi(3)(4 + 3)
\]
Calculating inside the parentheses first:
\[
h + r = 4 + 3 = 7
\]
Now plug that back into the formula:
\[
SA = 2\pi(3)(7)
\]
\[
SA = 2\pi(21)
\]
\[
SA = 42\pi
\]
Now we can use an approximate value for \( \pi \) (approximately 3.14) to calculate the numerical value:
\[
SA \approx 42 \times 3.14 = 131.88 \text{ yds}^2
\]
This is not in your options. Let's calculate it in a detailed way:
1. The lateral surface area of the cylinder is given by:
\[
LSA = 2\pi rh = 2\pi(3)(4) = 24\pi
\]
which is approximately \( 24 \times 3.14 \approx 75.36 \text{ yds}^2 \).
2. The area of the top and bottom circles is:
\[
A_{circles} = 2\pi r^2 = 2\pi(3^2) = 2\pi(9) = 18\pi
\]
which is approximately \( 18 \times 3.14 \approx 56.52 \text{ yds}^2 \).
3. Now, add both areas together to get the total surface area:
\[
SA = LSA + A_{circles} = 24\pi + 18\pi = 42\pi \approx 131.88 \text{ yds}^2
\]
Since we have calculated both lateral and circular bases separately and reached the total area which is not there in the provided options, I will summarize. The provided multiple choice answers do not include the correct calculated surface area for the cylinder of radius 3 yards and height 4 yards.
Assuming a different interpretation or error since it appears that none of your choices directly match the surface area calculated, it’s crucial we verify if the question or specifications are correct or reach the right approximations. If truly these were the only options presented, please reconfirm the numbers, or look for potential discrepancies in the specifications.