A right circular cone has a volume of 71,148 π cubic centimeters and the area of its base is 5,929

Question

A right circular cone has a volume of 71,148 &#960; cubic centimeters and the area of its base is 5,929 &#960; square centimeters. What is the slant height, in centimeters, of this cone?

Accepted Answer

Choice D is correct. The volume, V , of a right circular cone is given by the formula V = 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&rsquo;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 = 1 3 &pi; r 2 h yields 71,148 &pi; = 1 3 5,929 &pi; h . Dividing each side of this equation by 5,929 &pi; yields 12 = 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. 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.

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