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Geometry and Trigonometry / Area and volume Difficulty: Hard

Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is $58 square centimeters left parenthesis cm squared right parenthesis$ , and the surface area of right rectangular prism Y is $1,450 cm squared$ . The volume of right rectangular prism Y is $1,250 cubic centimeters left parenthesis cm cubed right parenthesis$ . What is the sum of the volumes, $in {cm}^{3}$ , of right rectangular prism X and right rectangular prism Y?

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Explanation

The correct answer is $1,260$ . Since it's given that prisms X and Y are similar, all the linear measurements of prism Y are $k$ times the respective linear measurements of prism X, where $k$ is a positive constant. Therefore, the surface area of prism Y is $k^{2}$ times the surface area of prism X and the volume of prism Y is $k^{3}$ times the volume of prism X. It's given that the surface area of prism Y is $1,450 c m squared$ , and the surface area of prism X is $58 c m squared$ , which implies that $1,450 equals 58 k squared$ . Dividing both sides of this equation by $58$ yields $\frac{1,450}{58} = k^{2}$ , or $k squared equals 25$ . Since $k$ is a positive constant, $k equals 5$ . It's given that the volume of prism Y is $1,250 c m cubed$ . Therefore, the volume of prism X is equal to $StartFraction 1,250 Over k cubed EndFraction c m cubed$ , which is equivalent to $StartFraction 1,250 Over 5 cubed EndFraction c m cubed$ , or $10 c m cubed$ . Thus, the sum of the volumes, in ${cm}^{3}$ , of the two prisms is $1,250 plus 10$ , or $1,260$ .