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

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



Answer :

Sure, let's solve the equation [tex]\( x = \frac{2}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex] step-by-step:

1. Start with the given equation:

[tex]\[ x = \frac{2}{3} \pi r^3 \][/tex]

2. Isolate [tex]\( r^3 \)[/tex]:

To isolate [tex]\( r^3 \)[/tex], multiply both sides of the equation by the reciprocal of the coefficient of [tex]\( r^3 \)[/tex], which is [tex]\(\frac{3}{2 \pi}\)[/tex]:

[tex]\[ x = \frac{2}{3} \pi r^3 \implies x \cdot \frac{3}{2 \pi} = \left( \frac{2}{3} \pi r^3 \right) \cdot \frac{3}{2 \pi} \][/tex]

The right side simplifies to [tex]\( r^3 \)[/tex]:

[tex]\[ \frac{3x}{2\pi} = r^3 \][/tex]

3. Solve for [tex]\( r \)[/tex]:

To solve for [tex]\( r \)[/tex], take the cube root of both sides of the equation:

[tex]\[ r = \sqrt[3]{\frac{3x}{2\pi}} \][/tex]

4. Determine the correct option:

We need to compare this derived expression with the given options:

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

Clearly, Option A matches our derived expression exactly.

Therefore, the correct solution is:

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