The formula for the volume of a sphere is [tex]V=\frac{4}{3} \pi r^3[/tex]. What is the formula solved for [tex]r[/tex]?

A. [tex]r=3 V+4 \pi[/tex]
B. [tex]r=\sqrt[3]{3 V-4 \pi}[/tex]
C. [tex]r=\sqrt[3]{\frac{3 V}{4 \pi}}[/tex]
D. [tex]r^3=3 V-4 \pi[/tex]



Answer :

To solve for [tex]\( r \)[/tex] from the formula for the volume of a sphere, [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex], follow these steps:

1. Start with the given equation:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]

2. Isolate [tex]\( r^3 \)[/tex]:
To do this, multiply both sides of the equation by [tex]\( \frac{3}{4\pi} \)[/tex] to cancel out the coefficient on the right side.
[tex]\[ \frac{3V}{4\pi} = r^3 \][/tex]

3. Solve for [tex]\( r \)[/tex]:
Take the cube root of both sides to solve for [tex]\( r \)[/tex].
[tex]\[ r = \sqrt[3]{\frac{3V}{4\pi}} \][/tex]

Thus, the formula solved for [tex]\( r \)[/tex] is:
[tex]\[ r = \sqrt[3]{\frac{3V}{4\pi}} \][/tex]

Therefore, the correct choice from the options given is:
[tex]\[ r = \sqrt[3]{\frac{3V}{4\pi}} \][/tex]