To find the radius of the cone, we will use the formula for the volume of a cone. The formula for the volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume of the cone.
- [tex]\( r \)[/tex] is the radius of the base of the cone.
- [tex]\( h \)[/tex] is the height of the cone.
We are given:
- The volume [tex]\( V \)[/tex] is [tex]\( 16 \pi \)[/tex] cubic inches.
- The height [tex]\( h \)[/tex] is 12 inches.
Let's substitute these values into the formula and solve for [tex]\( r \)[/tex]:
[tex]\[ 16 \pi = \frac{1}{3} \pi r^2 \times 12 \][/tex]
First, simplify the right-hand side:
[tex]\[ 16 \pi = \frac{1}{3} \pi r^2 \times 12 \][/tex]
[tex]\[ 16 \pi = 4 \pi r^2 \][/tex]
Next, divide both sides of the equation by [tex]\( \pi \)[/tex] to cancel out the [tex]\( \pi \)[/tex] terms:
[tex]\[ 16 = 4 r^2 \][/tex]
Now, divide both sides by 4 to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ \frac{16}{4} = r^2 \][/tex]
[tex]\[ 4 = r^2 \][/tex]
Finally, take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{4} \][/tex]
[tex]\[ r = 2 \][/tex]
So, the radius of the cone is [tex]\( 2 \)[/tex] inches. Therefore, the correct choice is:
[tex]\[ \boxed{2 \text{ in.}} \][/tex]
