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Advanced Math / Nonlinear functions Difficulty: Hard

The surface area of a cube is $6 times, open parenthesis, the fraction a, over 4, close parenthesis, squared$ , where a is a positive constant. Which of the following gives the perimeter of one face of the cube?

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Explanation

Choice B is correct. A cube has 6 faces of equal area, so if the total surface area of a cube is $6 times, open parenthesis, a, over 4, close parenthesis, squared$ , then the area of one face is $open parenthesis, a, over 4, close parenthesis, squared$ . Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is $open parenthesis, a, over 4, close parenthesis, squared$ , then the length of one edge of the cube is $a, over 4$ . Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is $4 times the fraction a, over 4, equals a$ .
Choice A is incorrect because if the perimeter of one face of the cube is $a, over 4$ , then the total surface area of the cube is $6 times, open parenthesis, the fraction with numerator a, over 4, and denominator 4, close parenthesis, squared, equals, 6 times, open parenthesis, a, over 16, close parenthesis, squared$ , which is not $6 times, open parenthesis, a, over 4, close parenthesis, squared$ . Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is $6 times, open parenthesis, the fraction 4 a, over 4, close parenthesis, squared, equals 6 a, squared$ , which is not $6 times, open parenthesis, a, over 4, close parenthesis, squared$ . Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface area of the cube is $6 times, open parenthesis, the fraction 6 a, over 4, close parenthesis, squared, equals, 6 times, open parenthesis, the fraction 3 a, over 2, close parenthesis, squared$ , which is not $6 times, open parenthesis, a, over 4, close parenthesis, squared$ .