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Geometry and Trigonometry Difficulty: Hard

A right circular cone has a volume of $71,148 pi$ cubic centimeters and the area of its base is $5,929 pi$ square centimeters. What is the slant height, in centimeters, of this cone?

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Explanation

Choice D is correct. The volume, $V$ , of a right circular cone is given by the formula $V = \frac{1}{3} π r^{2} h$ , where $π r^{2}$ 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 $π r^{2}$ in the formula $V = \frac{1}{3} π r^{2} h$ yields $71,148 pi equals left parenthesis one third right parenthesis left parenthesis 5,929 pi right parenthesis left parenthesis h right parenthesis$ . Dividing each side of this equation by $5,929 pi$ yields $12 = \frac{h}{3}$ . Multiplying each side of this equation by $3$ yields $36 equals h$ . 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 $π r^{2}$ , it follows that $5,929 pi equals pi r squared$ . Dividing both sides of this equation by $π$ yields $5,929 equals r squared$ . 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 $7,225 equals s squared$ . Taking the positive square root of both sides of this equation yields $85 equals s$ . Therefore, the slant height of the cone is $85$ centimeters.

Choice A is incorrect. This is one-third of the height, in centimeters, not the slant height, in centimeters, of this cone.

Choice B is incorrect. This is the height, in centimeters, not the slant height, in centimeters, of this cone.

Choice C is incorrect. This is the radius, in centimeters, of the base, not the slant height, in centimeters, of this cone.