A solid sphere is cut into 6 equal wedges. The volume of each wedge is [tex]\( V=\frac{2}{9} \pi r^3 \)[/tex]. Solve the formula for [tex]\( r \)[/tex].

A. [tex]\( r=\sqrt[3]{\frac{9 V}{2 \pi}} \)[/tex]

B. [tex]\( r=\sqrt[3]{\frac{2 \pi}{9 V}} \)[/tex]

C. [tex]\( r=\sqrt[3]{9 V(2 \pi)} \)[/tex]

D. [tex]\( r=\sqrt[3]{9 V-2 \pi} \)[/tex]



Answer :

To find the radius [tex]\( r \)[/tex] when the volume of each wedge is given by [tex]\( V = \frac{2}{9} \pi r^3 \)[/tex], we need to isolate [tex]\( r \)[/tex] in this equation. Let's solve it step-by-step:

1. Start with the given formula for the volume of each wedge:
[tex]\[ V = \frac{2}{9} \pi r^3 \][/tex]

2. Rearrange the equation to solve for [tex]\( r^3 \)[/tex]. To do this, multiply both sides of the equation by [tex]\(\frac{9}{2\pi}\)[/tex] to isolate [tex]\( r^3 \)[/tex] on one side:
[tex]\[ r^3 = \frac{9V}{2\pi} \][/tex]

3. Take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt[3]{\frac{9V}{2\pi}} \][/tex]

Therefore, the correct answer is:

A. [tex]\( r = \sqrt[3]{\frac{9V}{2\pi}} \)[/tex]