Question

the volume of a spikere of radius r is v = 4/3 πr³. complete parts (a)-(c)
a. use the formula to find r in terms of v. remove all radicals from the denominator.
r = □ (type an exact answer, using π as needed.)

Explanation:

Step1: Write the formula for the volume of a sphere

The volume formula of a sphere is \(V=\frac{4}{3}\pi r^{3}\).

Step2: Solve the formula for \(r\)

First, multiply both sides of the equation \(V = \frac{4}{3}\pi r^{3}\) by \(\frac{3}{4\pi}\):
\(r^{3}=\frac{3V}{4\pi}\).
Then, take the cube - root of both sides:
\(r=\sqrt[3]{\frac{3V}{4\pi}}\).
To rationalize the denominator, we can rewrite it as \(r=\frac{\sqrt[3]{3V\times4^{2}\pi^{2}}}{4\pi}\) (by multiplying the numerator and denominator by \(\sqrt[3]{4^{2}\pi^{2}}\) to make the denominator a perfect cube).

Answer:

\(r = \frac{\sqrt[3]{12V\pi^{2}}}{4\pi}\)