Solve the equation [tex]x = \frac{2}{3} \pi r^3[/tex] for [tex]r[/tex].

A. [tex]r = \sqrt[3]{\frac{3x - 2\pi}{\pi}}[/tex]
B. [tex]r = \sqrt[3]{\frac{9\pi}{2\pi}}[/tex]
C. [tex]r = \sqrt[3]{3x(2\pi)}[/tex]
D. [tex]r = \sqrt[3]{\frac{9\pi}{3x}}[/tex]



Answer :

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]