To determine the radius of a sphere that has the same surface area as a cylinder with a height of 4 meters and a radius of 1.5 meters, we will follow these steps:
1. Compute the Surface Area of the Cylinder:
The surface area [tex]\( S \)[/tex] of a cylinder is given by the formula:
[tex]\[
S_{\text{cylinder}} = 2\pi r (r + h)
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
Plugging in the given values [tex]\( r = 1.5 \, \text{meters} \)[/tex] and [tex]\( h = 4 \, \text{meters} \)[/tex]:
2. Calculate the Surface Area:
[tex]\[
S_{\text{cylinder}} = 2 \pi (1.5) (1.5 + 4)
\][/tex]
[tex]\[
= 2 \pi (1.5) (5.5)
\][/tex]
[tex]\[
\approx 2 \times 3.1416 \times 1.5 \times 5.5
\][/tex]
[tex]\[
\approx 51.836 \, \text{square meters}
\][/tex]
3. Surface Area of the Sphere:
The surface area [tex]\( S \)[/tex] of a sphere is given by the formula:
[tex]\[
S_{\text{sphere}} = 4 \pi r^2
\][/tex]
We know the surface area of the sphere is equal to the surface area of the cylinder, so:
[tex]\[
4 \pi r_{\text{sphere}}^2 = 51.836
\][/tex]
4. Solve for the Radius of the Sphere:
Isolating [tex]\( r_{\text{sphere}} \)[/tex] we get:
[tex]\[
r_{\text{sphere}}^2 = \frac{51.836}{4 \pi}
\][/tex]
[tex]\[
r_{\text{sphere}}^2 = \frac{51.836}{12.5664}
\][/tex]
[tex]\[
r_{\text{sphere}}^2 \approx 4.125
\][/tex]
[tex]\[
r_{\text{sphere}} \approx \sqrt{4.125}
\][/tex]
[tex]\[
r_{\text{sphere}} \approx 2.031 \, \text{meters}
\][/tex]
5. Select the Closest Option:
The closest option to our calculated radius of about 2.0 meters is:
[tex]\[
\boxed{2.0 \, \text{m}}
\][/tex]
Therefore, the correct answer is:
C. [tex]\( 2.0 \, \text{m} \)[/tex]
