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Geometry and Trigonometry / Area and volume Difficulty: Hard

A right circular cone has a volume of $one third, pi$ cubic feet and a height of 9 feet. What is the radius, in feet, of the base of the cone?

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Explanation

Choice A is correct. The equation for the volume of a right circular cone is $V equals, one third pi r squared times h$ . It’s given that the volume of the right circular cone is $one third pi$ cubic feet and the height is 9 feet. Substituting these values for V and h, respectively, gives $one third pi equals, one third pi r squared, times 9$ . Dividing both sides of the equation by $one third pi$ gives $1 equals, r squared times 9$ . Dividing both sides of the equation by 9 gives $one ninth equals r squared$ . Taking the square root of both sides results in two possible values for the radius, $the square root of one ninth$ or $the negative of the square root of one ninth$ . Since the radius can’t have a negative value, that leaves $the square root of one ninth$ as the only possibility. Applying the quotient property of square roots, $the square root of the fraction a, over b, equals, the fraction the square root of a, over the square root of b$ , results in $r equals, the fraction the square root of 1 over the square root of 9$ , or $r equals one third$ .

Choices B and C are incorrect and may result from incorrectly evaluating $the square root of one ninth$ . Choice D is incorrect and may result from solving $r squared equals 9$ instead of $r squared equals one ninth$ .