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
.