1986 AIME Problems/Problem 14

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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

In the above diagram, we focus on the line that appears closest and is parallel to $BC$ . All the blue lines are perpendicular lines to $BC$ and their other points are on $AB$ , the main diagonal. The green lines are projections of the blue lines onto the bottom face; all of the green lines originate in the corner and reach out to $AC$ , and have the same lengths as their corresponding blue lines. So we want to find the shortest distance between $AC$ and that corner, which is $\frac {wl}{\sqrt {w^2 + l^2}}$ .

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 invert the above equations to get a system of linear equations in $\frac {1}{h^2}$ , $\frac {1}{l^2}$ , and $\frac {1}{w^2}$ : $\frac {1}{l^2} + \frac {1}{w^2} = \frac {45}{900}$ $\frac {1}{h^2} + \frac {1}{w^2} = \frac {13}{900}$ $\frac {1}{h^2} + \frac {1}{l^2} = \frac {40}{900}$

We see that $h = 15$ , $l = 5$ , $w = 10$ . Therefore $V = 5 \cdot 10 \cdot 15 = 750$ .

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

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1986 AIME Problems/Problem 14

From AoPSWiki

Problem

Solution

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