Answered

The volume of a cone is [tex]\(3 \pi x^3\)[/tex] cubic units, and its height is [tex]\(x\)[/tex] units.

Which expression represents the radius of the cone's base, in units?

A. [tex]\(3 x\)[/tex]
B. [tex]\(6 x\)[/tex]
C. [tex]\(3 \pi x^2\)[/tex]
D. [tex]\(9 \pi x^2\)[/tex]



Answer :

GinnyAnswer
Let's determine the radius of the base of the cone given the volume and the height.

### Step-by-Step Solution:

1. Given Information:
- Volume of the cone: [tex]\( V = 3 \pi x^3 \)[/tex] cubic units
- Height of the cone: [tex]\( h = x \)[/tex] units

2. Formula of the Volume of a Cone:
The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius of the base and [tex]\( h \)[/tex] is the height of the cone.

3. Substitute the Given Values:
- Substitute [tex]\( V = 3 \pi x^3 \)[/tex] and [tex]\( h = x \)[/tex] into the formula:
[tex]\[ 3 \pi x^3 = \frac{1}{3} \pi r^2 x \][/tex]

4. Simplify the Equation:
- Multiply both sides by 3 to eliminate the fraction:
[tex]\[ 9 \pi x^3 = \pi r^2 x \][/tex]

- Divide both sides by [tex]\( \pi \)[/tex] to cancel [tex]\( \pi \)[/tex]:
[tex]\[ 9 x^3 = r^2 x \][/tex]

5. Solve for [tex]\( r \)[/tex]:
- Divide both sides by [tex]\( x \)[/tex] (assuming [tex]\( x \neq 0 \)[/tex]):
[tex]\[ 9 x^2 = r^2 \][/tex]

- Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{9 x^2} \][/tex]

6. Simplify the Square Root:
[tex]\[ r = 3 x \][/tex]

### Conclusion:
The radius of the base of the cone is [tex]\( 3 x \)[/tex] units.

So, the correct choice is:
[tex]\[ 3 x \][/tex]

The answer is [tex]\( 3 x \)[/tex].

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