To solve the equation [tex]\( x = \frac{2}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex], follow these steps:
1. Isolate the term involving [tex]\( r^3 \)[/tex]:
Start with the given equation:
[tex]\[
x = \frac{2}{3} \pi r^3
\][/tex]
To isolate [tex]\( r^3 \)[/tex], multiply both sides of the equation by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x \cdot \frac{3}{2} = \pi r^3
\][/tex]
Simplifying the left-hand side:
[tex]\[
\frac{3x}{2} = \pi r^3
\][/tex]
2. Solve for [tex]\( r^3 \)[/tex]:
Next, divide both sides by [tex]\(\pi\)[/tex] to isolate [tex]\( r^3 \)[/tex]:
[tex]\[
\frac{3x}{2\pi} = r^3
\][/tex]
3. Solve for [tex]\( r \)[/tex]:
Finally, take the cube root of both sides of the equation to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt[3]{\frac{3x}{2\pi}}
\][/tex]
Thus, the correct solution for [tex]\( r \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[
r = \sqrt[3]{\frac{3x}{2\pi}}
\][/tex]
Looking at the given answer choices, this corresponds to option C.
Therefore, the correct answer is:
C. [tex]\( r = \sqrt[3]{\frac{3x}{2\pi}} \)[/tex]
