Question

Find the surface area of the cylinder.
Imagine a small line on top of the cylinder labelled "4m"
The side of the cylinder is labelled "10m"
A. 251.33 m^2
B. 351.86 m^2
C. 100.53 m^2
D. 121.34 m^2

Answer 1

To find the surface area of a cylinder, we use the formula:

\[
SA = 2\pi r(h + r)
\]

where:
- \( SA \) is the surface area,
- \( r \) is the radius,
- \( h \) is the height of the cylinder.

From the information given:
- The height \( h \) of the cylinder is \( 10 \) m.
- The diameter of the cylinder is \( 4 \) m, which means the radius \( r \) is half of that:

\[
r = \frac{4}{2} = 2 \text{ m}
\]

Now, we can substitute the values into the surface area formula.

1. Calculate \( h + r \):
\[
h + r = 10 + 2 = 12 \text{ m}
\]

2. Substitute values into the surface area formula:
\[
SA = 2\pi (2)(12)
\]
\[
SA = 2\pi (24)
\]
\[
SA = 48\pi
\]

3. Now, use an approximate value of \( \pi \approx 3.14 \) to find the surface area:
\[
SA \approx 48 \times 3.14 \approx 150.72 \text{ m}^2
\]

However, I made a mistake. The correct surface area calculation includes both the lateral area and the area of the two circles on the top and bottom of the cylinder. Let's break it down:

- The lateral surface area \( A_{lateral} = 2\pi rh \):
\[
A_{lateral} = 2\pi (2)(10) = 40\pi
\]

- The area of the top and bottom circles \( A_{circles} = 2\pi r^2 \):
\[
A_{circles} = 2\pi (2^2) = 8\pi
\]

Now, combine these two areas to find the total surface area:
\[
SA = A_{lateral} + A_{circles} = 40\pi + 8\pi = 48\pi
\]

Again, when we calculate:
\[
SA \approx 48 \times 3.14 \approx 150.72 \text{ m}^2
\]

This value does not match any of the given options, which seems incorrect based on our initial misunderstanding.

Let’s check calculations one more time:

1. **Lateral Area**:
\( 2\pi rh = 2\pi (2)(10) = 40\pi \approx 125.66 \) m².
2. **Area of the two bases**:
\( 2\pi r^2 = 2\pi (2^2) = 8\pi \approx 25.13 \) m².

**Total Surface Area**:
\[
SA = 40\pi + 8\pi = 48\pi \approx 150.72 \text{ m}^2.
\]

Finding the same thing with the options leads to clarifying values or any misinterpretation. Given the surface area calculations above, none of the options provided fit.

Please verify the values for height and diameter once more, or if additional information/context from your problem is provided, stay transparent.