SAT Question dc71597b: Area and Volume answer and explanation with Desmos solution

Choice A is correct. The equation for the volume of a right circular cone is $V=\frac{1}{3}\pi r^{2}h$. It's given that the volume of the right circular cone is $\frac{1}{3}\pi$ cubic feet and the height is $9$ feet. Substituting these values for $V$ and $h$, respectively, gives $\frac{1}{3}\pi =\frac{1}{3}\pi r^2\times 9$. Dividing both sides of the equation by $\frac{1}{3}\pi$ gives $1=r^2\times 9$. Dividing both sides of the equation by $9$ gives $\frac{1}{9}=r^2$. Taking the square root of both sides results in two possible values for the radius, $\sqrt{\frac{1}{9}}$ or $-\sqrt{\frac{1}{9}}$. Since the radius can't have a negative value, that leaves $\sqrt{\frac{1}{9}}$ as the only possibility. Applying the quotient property of square roots, $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, results in $r=\frac{\sqrt{1}}{\sqrt{9}}$, or $r=\frac{1}{3}$.