To find the diameter of a sphere when the surface area is given as [tex]\(50.27 \, m^2\)[/tex], follow these steps:
1. Understand the formula for the surface area of a sphere:
[tex]\[
A = 4 \pi r^2
\][/tex]
where [tex]\(A\)[/tex] is the surface area and [tex]\(r\)[/tex] is the radius of the sphere.
2. Rearrange the formula to solve for the radius ([tex]\(r\)[/tex]):
[tex]\[
r^2 = \frac{A}{4 \pi}
\][/tex]
[tex]\[
r = \sqrt{\frac{A}{4 \pi}}
\][/tex]
3. Substitute the given surface area [tex]\(A = 50.27 \, m^2\)[/tex] into the formula:
[tex]\[
r = \sqrt{\frac{50.27}{4 \pi}}
\][/tex]
4. Calculate the value inside the square root and then take the square root to find [tex]\(r\)[/tex]:
- The calculation step isn't shown here, but we know the numerical result for [tex]\(r\)[/tex] is approximately [tex]\(2.0000907163368256 \, m\)[/tex].
5. Determine the diameter of the sphere:
- The diameter ([tex]\(d\)[/tex]) of a sphere is twice the radius ([tex]\(r\)[/tex]):
[tex]\[
d = 2r
\][/tex]
6. Substitute the radius back into the diameter formula:
[tex]\[
d = 2 \times 2.0000907163368256 \, m
\][/tex]
[tex]\[
d = 4.000181432673651 \, m
\][/tex]
Since the calculated diameter is very close to [tex]\(4 \, m\)[/tex], the closest answer from the provided options is:
[tex]\[
\boxed{4 \, m}
\][/tex]
So, the diameter of the sphere, given the surface area of [tex]\(50.27 \, m^2\)[/tex], is approximately [tex]\(4 \, m\)[/tex].
