Question

solving cubic and cube root equations
the volume ( v ) of a sphere can be found with the formula ( v = \frac{4}{3}pi r^3 ). if a weather balloon is inflated to a volume of ( 36pi ) cubic feet, what is the radius?
( r = square ) ft

Explanation:

Step1: Identify the volume formula

The volume \( V \) of a sphere is given by \( V=\frac{4}{3}\pi r^{3} \). We know the volume \( V = 36\pi \) (assuming the volume is \( 36\pi \) cubic feet, since the problem mentions "a volume of \( 36\pi \) cubic feet" - maybe a typo in the original, but this is a common problem setup).

Step2: Substitute \( V \) into the formula

Substitute \( V = 36\pi \) into \( V=\frac{4}{3}\pi r^{3} \):
\[
36\pi=\frac{4}{3}\pi r^{3}
\]

Step3: Divide both sides by \( \pi \)

Divide both sides of the equation by \( \pi \) to simplify:
\[
36=\frac{4}{3}r^{3}
\]

Step4: Multiply both sides by \( \frac{3}{4} \)

Multiply both sides by \( \frac{3}{4} \) to isolate \( r^{3} \):
\[
r^{3}=36\times\frac{3}{4}=27
\]

Step5: Take the cube root of both sides

Take the cube root of both sides to solve for \( r \):
\[
r = \sqrt[3]{27}=3
\]

Answer:

\( 3 \)