Question

the volume of a sphere of radius r is $v=\frac{4}{3}pi r^{3}$. complete parts (a)-(c)

a. use the formula to find r in terms of v. remove all radicals from the denominator

$r = square$

(type an exact answer, using $pi$ as needed)

Explanation:

Step1: Solve the volume formula for \( r \)

Given \( V=\frac{4}{3}\pi r^{3} \), first multiply both sides by \( \frac{3}{4\pi} \) to isolate \( r^{3} \).

$$ r^{3}=\frac{3V}{4\pi} $$

Step2: Take the cube - root of both sides

$$ r = \sqrt[3]{\frac{3V}{4\pi}}=\sqrt[3]{\frac{3V}{4\pi}}\times\frac{\sqrt[3]{2\pi^{2}}}{\sqrt[3]{2\pi^{2}}}=\sqrt[3]{\frac{6V\pi^{2}}{8\pi^{3}}}=\frac{\sqrt[3]{6V\pi^{2}}}{2\pi} $$

Answer:

\( r=\frac{\sqrt[3]{6V\pi^{2}}}{2\pi} \)