SAT Question 1be8b6b2: Area and Volume answer and explanation with Desmos solution

Choice D is correct. The volume, $V$, of a right circular cone is given by the formula $V=\frac{1}{3}\pi r^{2}h$, where $\pi r^2$ is the area of the circular base of the cone and $h$ is the height. It's given that this right circular cone has a volume of $71,148\pi$ cubic centimeters and the area of its base is $5,929\pi$ square centimeters. Substituting $71,148\pi$ for $V$ and $5,929\pi$ for $\pi r^2$ in the formula $V=\frac{1}{3}\pi r^{2}h$ yields $71,148\pi =\left(\frac{1}{3}\right)\left(5,929\pi \right)\left(h\right)$. Dividing each side of this equation by $5,929\pi$ yields $12=\frac{h}{3}$. Multiplying each side of this equation by $3$ yields $36=h$. Let $s$ represent the slant height, in centimeters, of this cone. A right triangle is formed by the radius, $r$, height, $h$, and slant height, $s$, of this cone, where $r$ and $h$ are the legs of the triangle and $s$ is the hypotenuse. Using the Pythagorean theorem, the equation $r^2+h^2=s^2$ represents this relationship. Because $5,929\pi$ is the area of the base and the area of the base is $\pi r^2$, it follows that $5,929\pi =\pi r^2$. Dividing both sides of this equation by $\pi$ yields $5,929=r^2$. Substituting $5,929$ for $r^2$ and $36$ for $h$ in the equation $r^2+h^2=s^2$ yields $5,929+36^2=s^2$, which is equivalent to $5,929+1,296=s^2$, or $7,225=s^2$. Taking the positive square root of both sides of this equation yields $85=s$. Therefore, the slant height of the cone is $85$ centimeters.