To find the radius of a hemisphere with a volume of [tex]\(80 \, \text{cm}^3\)[/tex], we need to follow a series of steps involving the formula for the volume of a sphere. Here is the detailed, step-by-step solution:
1. Volume Relationship Between Hemisphere and Sphere:
The volume [tex]\(V_{\text{hemisphere}}\)[/tex] of a hemisphere is half of the volume [tex]\(V_{\text{sphere}}\)[/tex] of a full sphere. Given the volume of the hemisphere is [tex]\(80 \, \text{cm}^3\)[/tex]:
[tex]\[
V_{\text{sphere}} = 2 \times V_{\text{hemisphere}}
\][/tex]
Substituting the given volume:
[tex]\[
V_{\text{sphere}} = 2 \times 80 \, \text{cm}^3 = 160 \, \text{cm}^3
\][/tex]
2. Formula for the Volume of a Sphere:
The volume of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
where [tex]\(r\)[/tex] is the radius of the sphere. Given the volume [tex]\(V_{\text{sphere}} = 160 \, \text{cm}^3\)[/tex], we can set up the equation:
[tex]\[
160 = \frac{4}{3} \pi r^3
\][/tex]
3. Solving for the Radius:
We need to solve for [tex]\(r\)[/tex]. Start by isolating [tex]\(r^3\)[/tex]. Multiply both sides of the equation by [tex]\(\frac{3}{4 \pi}\)[/tex]:
[tex]\[
r^3 = 160 \times \frac{3}{4 \pi} = \frac{480}{4 \pi} = \frac{120}{\pi}
\][/tex]
4. Cube Root to Find the Radius:
To find [tex]\(r\)[/tex], we need to take the cube root of [tex]\(\frac{120}{\pi}\)[/tex]:
[tex]\[
r = \sqrt[3]{\frac{120}{\pi}}
\][/tex]
5. Simplification and Numerical Solution:
Evaluating the above expression numerically, we find:
[tex]\[
r \approx 3.367 \, \text{cm}
\][/tex]
So, the radius of the hemisphere is approximately [tex]\(3.367 \, \text{cm}\)[/tex].
