Question

the volume of a cone is $3pi x^3$ cubic units and its height is $x$ units. which expression represents the radius of the cones base, in units? $3x$ $3pi x^2$ $6x$ $9pi x^2$

Explanation:

Step1: Recall the volume formula of a cone

The volume \( V \) of a cone is given by the formula \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius of the base and \( h \) is the height of the cone.

Step2: Substitute the given values into the formula

We know that \( V = 3\pi x^{3} \) and \( h=x \). Substituting these values into the volume formula, we get:

$$ 3\pi x^{3}=\frac{1}{3}\pi r^{2}(x) $$

Step3: Solve for \( r^{2} \)

First, multiply both sides of the equation by 3 to eliminate the fraction on the right - hand side:

$$ 3\times(3\pi x^{3})=\pi r^{2}(x) $$

$$ 9\pi x^{3}=\pi r^{2}x $$

Then, divide both sides of the equation by \( \pi x \) (assuming \( x
eq0 \), and since we are dealing with a geometric figure, \( x \) represents a length and is non - zero).

$$ \frac{9\pi x^{3}}{\pi x}=r^{2} $$

Simplify the left - hand side: \( \frac{9\pi x^{3}}{\pi x}=9x^{2} \), so \( r^{2}=9x^{2} \)

Step4: Solve for \( r \)

Take the square root of both sides. Since \( r \) represents the radius (a non - negative quantity), we have \( r = \sqrt{9x^{2}}=3x \) (we take the positive square root because radius cannot be negative).

Answer:

\( 3x \) (the first option)