by Manhattan Prep, Nov 23, 2010
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Question
A rectangular solid has side lengths 3, 4, and z, where z is an integer. The total surface area of this solid is the same as that of a cube with side length s, where s is an integer. If z is the lowest possible integer that fits these conditions, what is z?(A) 3
(B) 5
(C) 6
(D) 7
(E) 9
Answer
The surface area of a rectangular solid is the total area of all of its faces. Since the side lengths are 3, 4, and z, there will be two faces with area 12 (= 34), two faces with area 3z, and two faces with area 4z. Thus, the total surface area of this solid will be 2 (12 + 3z + 4z) = 24 + 14z.
Meanwhile, since a cube has 6 faces, its surface area is 6 times the area of one face, or [pmath]6s^2[/pmath].
We now set these expressions equal to each other.
[pmath]24 + 14z = 6s^2[/pmath]
[pmath]12 + 7z = 3s^2[/pmath]
We need to find two integers, z and s, that fit this equation, and z needs to be its lowest possible value. At this point, we should switch to testing numbersalthough we can save ourselves some work if we notice that z must be a multiple of 3. (Why? Notice that the right side, [pmath]3s^2[/pmath] , is a multiple of 3, and on the left side, 12 is a multiple of 3. That means 7z must be a multiple of 3, and since 7 is not a multiple of 3 itself, z must be a multiple of 3.) Also, z must be a positive integer (otherwise, the rectangular solid would not be a solid), so we can start with z = 3 and check.
If [pmath]z = 3, 12 + 7z = 12 + 21 = 33 = 3s^2[/pmath], meaning that [pmath]s^2 = 11[/pmath], but 11 is not a perfect square. No good.
If If [pmath]z = 6, 12 + 7z = 12 + 42 = 54 = 3s^2[/pmath], meaning that [pmath]s^2 = 18[/pmath], but 18 is not a perfect square. No good.
If If [pmath]z = 9, 12 + 7z = 12 + 63 = 75 = 3^s[/pmath], meaning that [pmath]s^2 = 25[/pmath], so s = 5.This fits.
Since we are asked for s, we must pick 5. Be careful not to pick z (which would be answer E). The correct answer is (B).
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