Let's solve the equation [tex]\( x = \frac{2}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex] step-by-step:
1. First, we start with the given equation:
[tex]\[
x = \frac{2}{3} \pi r^3
\][/tex]
2. To isolate [tex]\( r^3 \)[/tex], we need to get rid of the fraction. Multiply both sides of the equation by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
\left( \frac{3}{2} \right) x = \pi r^3
\][/tex]
Simplifying, we have:
[tex]\[
\frac{3x}{2} = \pi r^3
\][/tex]
3. Next, divide both sides by [tex]\(\pi\)[/tex] to isolate [tex]\( r^3 \)[/tex]:
[tex]\[
\frac{3x}{2 \pi} = r^3
\][/tex]
4. Take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt[3]{\frac{3x}{2 \pi}}
\][/tex]
After working through these steps, we find that the solution for [tex]\( r \)[/tex] is:
[tex]\[
r = \sqrt[3]{\frac{3x}{2 \pi}}
\][/tex]
Checking the given options:
- A. [tex]\( r = \sqrt[3]{3 x - 2 \pi} \)[/tex]
- B. [tex]\( r = \sqrt[3]{\frac{9 \pi}{2 \pi}} \)[/tex]
- C. [tex]\( r = \sqrt[3]{3 x (2 \pi)} \)[/tex]
- D. [tex]\( r = \sqrt[3]{\frac{9 \pi}{3 x}} \)[/tex]
We see that none of these options match our derived solution [tex]\( r = \sqrt[3]{\frac{3x}{2 \pi}} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\text{None of the given options is correct}
\][/tex]
