Question
Find the surface area of the cylinder
Imagine an image of a cylinder. On its right side, is a label of "4 yd". Inside the cylinder, there's a blue line at the bottom connecting from the left side to the right side. On the right side above the line is a square shape, and the line is labelled "3 yd"
A. 64 cm^2
B. 16 cm^2
C. 84 cm^2
D. 48 cm^2
Imagine an image of a cylinder. On its right side, is a label of "4 yd". Inside the cylinder, there's a blue line at the bottom connecting from the left side to the right side. On the right side above the line is a square shape, and the line is labelled "3 yd"
A. 64 cm^2
B. 16 cm^2
C. 84 cm^2
D. 48 cm^2
Answers
GPT-4o mini
To calculate the surface area of a cylinder, we use the formula:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi r h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
From your description:
- The label "4 yd" suggests that this could be the height of the cylinder (h = 4 yd).
- The "3 yd" label likely indicates the diameter of the base, meaning the radius \( r = \frac{3}{2} \) yd = 1.5 yd.
Now we can substitute \( r \) and \( h \) into the surface area formula:
1. Convert all measurements to the same unit (since question options are in cm², we can convert yards to centimeters):
- \( 1 \text{ yd} = 91.44 \text{ cm} \)
- Radius: \( r = 1.5 \text{ yd} = 1.5 \times 91.44 \text{ cm} \approx 137.16 \text{ cm} \)
- Height: \( h = 4 \text{ yd} = 4 \times 91.44 \text{ cm} \approx 365.76 \text{ cm} \)
2. Calculate the surface area:
- First find \( r^2 \):
\[
r^2 \approx (137.16 \text{ cm})^2 \approx 18853.86 \text{ cm}^2
\]
- Now plug in \( r \) and \( h \):
\[
\text{Surface Area} = 2\pi (137.16)^2 + 2\pi (137.16)(365.76)
\]
- Calculate each term:
\[
2\pi r^2 \approx 2 \times 3.14 \times 18853.86 \approx 11883.25 \text{ cm}^2
\]
\[
2\pi rh \approx 2 \times 3.14 \times 137.16 \times 365.76 \approx 31780.80 \text{ cm}^2
\]
3. Combine these:
\[
\text{Total Surface Area} \approx 11883.25 + 31780.80 \approx 43664.05 \text{ cm}^2
\]
This value does not appear in any of the answer options, implying a miscommunication or misinterpretation of the dimensions. However, based on the description provided (the diameter of 3 yards seems incorrect). Please recheck the question and the measurements noted. The options provided are likely based on a different set of dimensions or numbers.
\[
\text{Surface Area} = 2\pi r^2 + 2\pi r h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
From your description:
- The label "4 yd" suggests that this could be the height of the cylinder (h = 4 yd).
- The "3 yd" label likely indicates the diameter of the base, meaning the radius \( r = \frac{3}{2} \) yd = 1.5 yd.
Now we can substitute \( r \) and \( h \) into the surface area formula:
1. Convert all measurements to the same unit (since question options are in cm², we can convert yards to centimeters):
- \( 1 \text{ yd} = 91.44 \text{ cm} \)
- Radius: \( r = 1.5 \text{ yd} = 1.5 \times 91.44 \text{ cm} \approx 137.16 \text{ cm} \)
- Height: \( h = 4 \text{ yd} = 4 \times 91.44 \text{ cm} \approx 365.76 \text{ cm} \)
2. Calculate the surface area:
- First find \( r^2 \):
\[
r^2 \approx (137.16 \text{ cm})^2 \approx 18853.86 \text{ cm}^2
\]
- Now plug in \( r \) and \( h \):
\[
\text{Surface Area} = 2\pi (137.16)^2 + 2\pi (137.16)(365.76)
\]
- Calculate each term:
\[
2\pi r^2 \approx 2 \times 3.14 \times 18853.86 \approx 11883.25 \text{ cm}^2
\]
\[
2\pi rh \approx 2 \times 3.14 \times 137.16 \times 365.76 \approx 31780.80 \text{ cm}^2
\]
3. Combine these:
\[
\text{Total Surface Area} \approx 11883.25 + 31780.80 \approx 43664.05 \text{ cm}^2
\]
This value does not appear in any of the answer options, implying a miscommunication or misinterpretation of the dimensions. However, based on the description provided (the diameter of 3 yards seems incorrect). Please recheck the question and the measurements noted. The options provided are likely based on a different set of dimensions or numbers.