To determine the diameter of a hemisphere given its total surface area, we can follow these steps:
1. Understand the formula for the total surface area of a hemisphere:
The total surface area [tex]\(A\)[/tex] of a hemisphere includes both the curved surface area and the base area. It is given by the formula:
[tex]\[
A = 3 \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius of the hemisphere.
2. Set up the equation with the given surface area:
We are given that the total surface area [tex]\(A\)[/tex] is [tex]\(36 \pi \, \text{cm}^2\)[/tex]. Therefore, we can write:
[tex]\[
3 \pi r^2 = 36 \pi
\][/tex]
3. Simplify the equation by canceling [tex]\(\pi\)[/tex]:
[tex]\[
3 r^2 = 36
\][/tex]
4. Solve for [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = \frac{36}{3} = 12
\][/tex]
5. Find the radius [tex]\(r\)[/tex] by taking the square root:
[tex]\[
r = \sqrt{12} \approx 3.464
\][/tex]
6. Determine the diameter:
The diameter [tex]\(d\)[/tex] is twice the radius. Thus, we calculate:
[tex]\[
d = 2r = 2 \times 3.464 \approx 6.928
\][/tex]
Hence, the diameter of the hemisphere is approximately [tex]\(6.928 \, \text{cm}\)[/tex].
