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Geometry and Trigonometry / Area and volume Difficulty: Hard

Two identical rectangular prisms each have a height of $90 centimeters left parenthesis cm right parenthesis$ . The base of each prism is a square, and the surface area of each prism is $K {cm}^{2}$ . If the prisms are glued together along a square base, the resulting prism has a surface area of $StartFraction 92 Over 47 EndFraction upper K cm squared$ . What is the side length, in $cm$ , of each square base?

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Explanation

Choice B is correct. Let $x$ represent the side length, in $cm$ , of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields $2 upper K minus 2 x squared cm squared$ . It’s given that this resulting surface area is equal to $StartFraction 92 Over 47 EndFraction upper K cm squared$ , so $2 K - 2 x^{2} = \frac{92}{47} K$ . Subtracting $StartFraction 92 Over 47 EndFraction upper K$ from both sides of this equation yields $2 K - \frac{92}{47} K - 2 x^{2} = 0$ . This equation can be rewritten by multiplying $2 upper K$ on the left-hand side by $StartFraction 47 Over 47 EndFraction$ , which yields $\frac{94}{47} K - \frac{92}{47} K - 2 x^{2} = 0$ , or $\frac{2}{47} K - 2 x^{2} = 0$ . Adding $2 x squared$ to both sides of this equation yields $two forty sevenths upper K equals 2 x squared$ . Multiplying both sides of this equation by $StartFraction 47 Over 2 EndFraction$ yields $upper K equals 47 x squared$ . The surface area $K$ , in $cm Superscript 2$ , of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is $90 cm$ and the side length of each square base is $x cm$ , it follows that the area of each square base is $x^{2} {cm}^{2}$ and the area of each lateral face is $90 x cm squared$ . Therefore, the surface area of each rectangular prism can be represented by the expression $2 x squared plus 4 left parenthesis 90 x right parenthesis$ , or $2 x squared plus 360 x$ . Substituting this expression for $K$ in the equation $upper K equals 47 x squared$ yields $2 x^{2} + 360 x = 47 x^{2}$ . Subtracting $2 x squared$ and $360 x$ from both sides of this equation yields $0 = 45 x^{2} - 360 x$ . Factoring $x$ from the right-hand side of this equation yields $0 = x (45 x - 360)$ . Applying the zero product property, it follows that $x equals 0$ and $45 x - 360 = 0$ . Adding $360$ to both sides of the equation $45 x - 360 = 0$ yields $45 x equals 360$ . Dividing both sides of this equation by $45$ yields $x equals 8$ . Since a side length of a rectangular prism can’t be $0$ , the length of each square base is $8$ $cm$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.