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Subject Test Math 1

Question 30 of 32

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SAT Subject Test

The front, side, and bottom faces of a rectangular solid have areas of 24 square centimeters, 8 square centimeters, and 3 square centimeters, respectively. What is the volume of the solid, in cubic centimeters?

Select an Answer

24

Correct Answer: 
Yes

96

Correct Answer: 
No

192

Correct Answer: 
No

288

Correct Answer: 
No

576

Correct Answer: 
No

The correct answer is A.

One way to determine the volume of the solid is to determine the length ell, width w, and height h of the solid, in centimeters, and then apply the formula V equals ell w h to compute the volume. Let ell, w, and h represent the length, width, and height, in centimeters, respectively, of the solid. The area of the front face of the solid is ell h equals 24 square centimeters, the area of the side face is w h equals 8 square centimeters, and the area of the bottom face is ell w equals 3 square centimeters. Elimination of h by using the first two equations gives fraction numerator ell h over denominator w h end fraction equals fraction numerator 24 over denominator 8 end fraction, which simplifies to fraction numerator ell over denominator w end fraction equals 3, or ell equals 3 w. Substitution of 3 w for ell in the third equation gives left parenthesis 3 w right parenthesis w equals 3, or 3 w to the power of 2 end exponent equals 3, so w equals 1 (since only positive values of w make sense as measurements of the length of any edge of a rectangular solid). Substitution of 1 for w in the equation w h equals 8 gives h equals 8, and substitution of 8 for h in the equation ell h equals 24 gives 8 ell equals 24, so ell equals 3. Therefore, the volume V of the solid, in cubic centimeters, is V equals left parenthesis 3 right parenthesis left parenthesis 1 right parenthesis left parenthesis 8 right parenthesis equals 24.

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 V equals ell w h gives Error converting from MathML to accessible text.. Therefore, in this case, V to the power of 2 end exponent equals left parenthesis 3 right parenthesis left parenthesis 24 right parenthesis left parenthesis 8 right parenthesis equals 576, so V equals square root of 576 end root equals 24. Note that it is not necessary to solve for the values of ell, w, and h.

Question Difficulty: 
hard