Question

A cone with height h and radius r has volume V = 1/3πr2h. If the cone has a height of 6 in. and volume V = 8πx2 + 24πx + 18π, what is its radius r in terms of x?

Answer 1

Answer:

C

Step-by-step explanation:

The volume for a cone is given by:

$\displaystyle V=\frac{1}{3}\pi r^2h$

For a given cone with a height of 6 inches, the volume is represented by:

$\displaystyle V=8\pi x^2+24\pi x+18\pi$

We want to find the radius r in terms of x.

Since the height is 6, this means that:

$\displaystyle V=\frac{1}{3}\pi r^2(6)=2\pi r^2$

By substitution:

$2\pi r^2=8\pi x^2+24\pi x+18\pi$

Divide both sides by 2π:

$\displaystyle r^2=\frac{8\pi x^2+24\pi x+18\pi}{2\pi}=4x^2+12x+9$

Factor the right. Notice that we have a perfect square trinomial*:

$4x^2+12x+9=(2x)^2+2(2x)(3)+(3)^2$

Factor:

$4x^2+12x+9=(2x+3)^2$

Therefore:

$r^2=(2x+3)^2$

Take the square root of both sides. The radius should be positive, so we only need to consider the positive case:

$r=\sqrt{(2x+3)^2}=2x+3$

And our answer is C!

*Note:

$a^2+2ab+b^2=(a+b)^2\\\\\text{In this case } a = 2x\text{ and } b =3$