Let's find the expression to determine [tex]\( r \)[/tex], the radius of the base of the cone, given certain values. The volume [tex]\( V \)[/tex] of a cone can be calculated using the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height,
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159).
We are provided with:
- The volume of the cone, [tex]\( V = 8 \pi \)[/tex] cubic centimeters,
- The height of the cone, [tex]\( h = 6 \)[/tex] cm.
The objective is to identify the correct expression for [tex]\( r \)[/tex].
To start, substitute the given values into the volume formula:
[tex]\[
8 \pi = \frac{1}{3} \pi r^2 \times 6
\][/tex]
Now, let's simplify the expression step-by-step:
1. Arrange the equation:
[tex]\[
8 \pi = \frac{1}{3} \pi \cdot r^2 \cdot 6
\][/tex]
2. Combine the π terms:
[tex]\[
8 \pi = \frac{1}{3} \pi \cdot 6 \cdot r^2
\][/tex]
3. Simplify the constants:
[tex]\[
8 \pi = 2 \pi r^2
\][/tex]
Therefore, the correct expression for finding [tex]\( r \)[/tex] is:
[tex]\[
8 \pi = \frac{1}{3} \pi \left( r^2 \right)(6)
\][/tex]
This matches precisely with one of the given answer choices:
[tex]\[
8 \pi=\frac{1}{3} \pi\left(r^2\right)(6)
\][/tex]
This is the correct answer.
