2007 iTest Problems/Problem 27

Problem

The face diagonal of a cube has length $4$. Find the value of n given that $n\sqrt2$ is the $\textit{volume}$ of the cube.

Solution

[asy]  draw((0,0)--(10,0)--(10,10)--(0,10)--(0,0)); draw((10,0)--(0,10)); label("$4$",(5,5),NE); label("$s$",(5,0),S); label("$s$",(0,5),W);  [/asy]

If the length of the face diagonal of the cube is $4$, then by using 45-45-90 triangles, the side length of the cube is $2 \sqrt{2}$. Thus, the volume of the cube is $(2 \sqrt{2})^3 = 16 \sqrt{2}$, so $n = \boxed{16}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 26
Followed by:
Problem 28
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