To solve for the radius [tex]\( r \)[/tex] in the volume formula for a sphere, we start with the given equation:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Our goal is to express [tex]\( r \)[/tex] in terms of [tex]\( V \)[/tex]. Let's follow the steps:
1. Isolate the term involving [tex]\( r \)[/tex]:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
To do this, we need to divide both sides of the equation by [tex]\(\frac{4}{3} \pi\)[/tex]:
[tex]\[ \frac{V}{\frac{4}{3} \pi} = r^3 \][/tex]
2. Simplify the fraction:
Dividing by a fraction is the same as multiplying by its reciprocal. So, we have:
[tex]\[ \frac{V \cdot 3}{4 \pi} = r^3 \][/tex]
Which simplifies to:
[tex]\[ \frac{3V}{4\pi} = r^3 \][/tex]
3. Solve for [tex]\( r \)[/tex] by taking the cube root of both sides:
To isolate [tex]\( r \)[/tex], we take the cube root of both sides of the equation:
[tex]\[ r = \sqrt[3]{\frac{3V}{4\pi}} \][/tex]
This gives us the primary real solution. However, the complete solution set when solving the equation [tex]\((\frac{4}{3} \pi r^3 = V\)[/tex] includes complex roots as well due to the nature of the cubic equation. Including all roots:
[tex]\[ r = \boxed{0.6203504908994 V^{1/3}, -0.3101752454497 V^{1/3} - 0.537239284369028 i V^{1/3}, -0.3101752454497 V^{1/3} + 0.537239284369028 i V^{1/3}} \][/tex]
These three values represent all possible solutions for [tex]\( r \)[/tex] in terms of [tex]\( V \)[/tex].
