Question 1015275: If a cube has an edge equal to the diagonal of another cube. Find the ratio of their volumes Found 2 solutions by rothauserc, Edwin McCravy:Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website! I assume your cube's diagonal is not a space diagonal
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let e1 be an edge of cube 1, then volume of cube 1 is e1^3
We know that e1 is the diagonal of cube 2, let e2 be an edge of cube
:
now use the Pythagorean Theorem
e1^2 = e2^2 + e2^2
2e2^2 = e1^2
e2 = e1/sqrt(2)
volume of cube 2 is e1^3 / (sqrt(2))^3 = e1^3 / (2sqrt(2))
:
the ratio(R) of their volumes is e1^3 / (e1^3 / (2sqrt(2)))
R = (e1^3 * 2sqrt(2)) / e1^3 = 2sqrt(2)
:
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The other tutor's solution uses the space diagonal, I did not use that since the problem stated "diagonal" only.
It should be clear that a cube has two types of diagonals, one is on a face and the other is inside - called a space diagonal.
The other tutor's solution above is incorrect.
Let x be the length of every edge of the smaller cube.
ΔABC and ΔBCE are right triangles. (They don't look it
because they're drawn in perspective to make a 2D drawing
look like 3D.) But ∠ACB is the right angle in right
triangle ΔABC, and ∠BCE is the right angle in right
triangle ΔBCE.
BC is the hypotenuse of right triangle ΔABC, and it is
also the bottom leg of triangle ABC,
Using the Pythagorean theorem on right triangle ΔABC:
Using the Pythagorean theorem on right triangle ΔABC:
Using the volume of a cube formula for the smaller cube:
Using the volume of a cube formula for the larger cube:
The ratio of the volume of the larger cube to the volume
of the smaller cube:
or the fraction:
to 1
So the volume of the larger cube is or about 5.2 times
the volume of the smaller cube.
Edwin
