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1986 AIME Problems/Problem 14

Revision as of 18:14, 29 September 2007 by Azjps (talk | contribs) (solution by minsoens)

Problem

The shortest distances between an interior diagonal of a rectangular parallelepiped, $P$, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.

Solution

If you draw a right triangle with the space diagonal as the hypotenuse, one side as a leg, and the corresponding face diagonal as the other leg, then you will notice that the minimum distance from the triangle to another side parallel to the first leg of the triangle is what the problem asks for.

In other words, draw a space diagonal of any face that shares a vertex with the space diagonal. It is only necessary to find the min. distance to another vertex of that face.

Using similar triangles and Pythagoras, the distance equation is: $\frac {xy}{\sqrt {x^2 + y^2}}$, where x and y are any of the sides.

So we have: \[\frac {lw}{\sqrt {l^2 + w^2}} = \frac {10}{\sqrt {5}}\] \[\frac {hw}{\sqrt {h^2 + w^2}} = \frac {30}{\sqrt {13}}\] \[\frac {hl}{\sqrt {h^2 + l^2}} = \frac {15}{\sqrt {10}}\]

Notice the familiar roots: $\sqrt {5}$, $\sqrt {13}$, $\sqrt {10}$, which are $\sqrt {1^2 + 2^2}$, $\sqrt {2^2 + 3^2}$, $\sqrt {1^2 + 3^2}$, respectively. (This would give us the guess that the sides are of the ratio 1:2:3, but let's provide the complete solution.)

\[\frac {l^2w^2}{l^2 + w^2} = \frac {1}{\frac {1}{l^2} + \frac {1}{w^2}} = 20\] \[\frac {h^2w^2}{h^2 + w^2} = \frac {1}{\frac {1}{h^2} + \frac {1}{w^2}} = \frac {900}{13}\] \[\frac {h^2l^2}{h^2 + l^2} = \frac {1}{\frac {1}{h^2} + \frac {1}{l^2}} = \frac {45}{2}\]

We solve the above equations for $\frac {1}{h^2}$, $\frac {1}{l^2}$, and $\frac {1}{w^2}$: \[\frac {1}{l^2} + \frac {1}{w^2} = \frac {45}{900}\] \[\frac {1}{h^2} + \frac {1}{w^2} = \frac {13}{900}\] \[\frac {1}{h^2} + \frac {1}{l^2} = \frac {40}{900}\]

$h = 15$, $l = 5$, $w = 10$, as expected, so $V = 5 \cdot 10 \cdot 15 = 750$.

See also

1986 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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