To determine the radius of a sphere given its volume, we use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Given that the volume [tex]\( V \)[/tex] is [tex]\( 36\pi \)[/tex] cubic units, we can set up the equation:
[tex]\[ 36\pi = \frac{4}{3} \pi r^3 \][/tex]
To solve for the radius [tex]\( r \)[/tex], we follow these steps:
1. Isolate the [tex]\( r^3 \)[/tex] term:
Divide both sides of the equation by [tex]\( \pi \)[/tex]:
[tex]\[ 36 = \frac{4}{3} r^3 \][/tex]
2. Eliminate the fraction:
Multiply both sides by [tex]\( 3 \)[/tex] to get rid of the denominator:
[tex]\[ 36 \times 3 = 4 r^3 \][/tex]
[tex]\[ 108 = 4 r^3 \][/tex]
3. Solve for [tex]\( r^3 \)[/tex]:
Divide both sides by [tex]\( 4 \)[/tex]:
[tex]\[ r^3 = \frac{108}{4} \][/tex]
[tex]\[ r^3 = 27 \][/tex]
4. Take the cube root:
Solving for [tex]\( r \)[/tex] involves taking the cube root of both sides:
[tex]\[ r = \sqrt[3]{27} \][/tex]
[tex]\[ r = 3 \][/tex]
Thus, the radius of the sphere is 3 units.
So, the correct option is:
O 3 units
