The front, side, and bottom faces of a rectangular solid have areas of square centimeters, square centimeters, and square centimeters, respectively. What is the volume of the solid, in cubic centimeters?
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The correct answer is A.
One way to determine the volume of the solid is to determine the length , width , and height of the solid, in centimeters, and then apply the formula to compute the volume. Let , , and represent the length, width, and height, in centimeters, respectively, of the solid. The area of the front face of the solid is square centimeters, the area of the side face is square centimeters, and the area of the bottom face is square centimeters. Elimination of by using the first two equations gives , which simplifies to , or . Substitution of for in the third equation gives , or , so (since only positive values of make sense as measurements of the length of any edge of a rectangular solid). Substitution of for in the equation gives , and substitution of for in the equation gives , so . Therefore, the volume of the solid, in cubic centimeters, is .
Alternatively, one can recognize that the square of the volume of a rectangular solid is the product of the areas of the front, side, and bottom faces of the solid. That is, squaring both sides of the formula gives . Therefore, in this case, , so . Note that it is not necessary to solve for the values of , , and .