ANSWER
[tex]r = x - \frac{3}{2} [/tex]
EXPLANATION
The volume of the cone is
[tex]V=8\pi \: {x}^{2} + 24\pi \: x + 18\pi[/tex]
The height of the is 6 inches.
We put the values into the volume of cone to get.
[tex]8\pi \: {x}^{2} + 24\pi \: x + 18\pi = \frac{1}{3} \times \pi \: {r}^{2} \times 6[/tex]
[tex]8\pi \: {x}^{2} + 24\pi \: x + 18\pi = 2 \pi \: {r}^{2} [/tex]
Divide through by 2π.
[tex]4 {x}^{2} + 12x + 9 = {r}^{2} [/tex]
The expression on the LHS is a perfect square trinomial.
[tex]( {x - \frac{3}{2} })^{2} = {r}^{2} [/tex]
[tex]r = x - \frac{3}{2} [/tex]
