To solve the equation [tex]\( x = \frac{5}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex], follow these steps:
1. Rewrite the equation for clarity:
[tex]\[
x = \frac{5}{3} \pi r^3
\][/tex]
2. Isolate [tex]\( r^3 \)[/tex]: Start by getting rid of the fraction. Multiply both sides of the equation by 3 to achieve this:
[tex]\[
3x = 5 \pi r^3
\][/tex]
3. Solve for [tex]\( r^3 \)[/tex]: Divide both sides by [tex]\( 5\pi \)[/tex]:
[tex]\[
r^3 = \frac{3x}{5\pi}
\][/tex]
4. Solve for [tex]\( r \)[/tex]: Take the cube root of both sides to isolate [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt[3]{\frac{3x}{5\pi}}
\][/tex]
Therefore, the solution to the equation [tex]\( x = \frac{5}{3} \pi r^3 \)[/tex] for [tex]\( r \)[/tex] is:
[tex]\[
\boxed{D: r = \sqrt[3]{\frac{3x}{5\pi}}}
\][/tex]
