Answer:
C
Step-by-step explanation:
The volume for a cone is given by:
[tex]\displaystyle V=\frac{1}{3}\pi r^2h[/tex]
For a given cone with a height of 6 inches, the volume is represented by:
[tex]\displaystyle V=8\pi x^2+24\pi x+18\pi[/tex]
We want to find the radius r in terms of x.
Since the height is 6, this means that:
[tex]\displaystyle V=\frac{1}{3}\pi r^2(6)=2\pi r^2[/tex]
By substitution:
[tex]2\pi r^2=8\pi x^2+24\pi x+18\pi[/tex]
Divide both sides by 2π:
[tex]\displaystyle r^2=\frac{8\pi x^2+24\pi x+18\pi}{2\pi}=4x^2+12x+9[/tex]
Factor the right. Notice that we have a perfect square trinomial*:
[tex]4x^2+12x+9=(2x)^2+2(2x)(3)+(3)^2[/tex]
Factor:
[tex]4x^2+12x+9=(2x+3)^2[/tex]
Therefore:
[tex]r^2=(2x+3)^2[/tex]
Take the square root of both sides. The radius should be positive, so we only need to consider the positive case:
[tex]r=\sqrt{(2x+3)^2}=2x+3[/tex]
And our answer is C!
*Note:
[tex]a^2+2ab+b^2=(a+b)^2\\\\\text{In this case } a = 2x\text{ and } b =3[/tex]
