Woodworking Again

Geometry Level 5

A person finds a piece of wood in a shape of irregular tetrahedron with edges of length in cm: $AB=7, AC=5, BC=6, AD=8, BD=10, CD=9$ .

Because the surface of the wooden block is damaged, the person decides to take: $1$ cm off face $ABC$ , $0.75$ cm off face $ACD$ , $0.5$ cm off face $BCD$ and $0.25$ cm off face $ABD$ . After doing so he notices further damage near the vertices of the tetrahedron so he decides to make a sphere shape. What is the radius $r$ of the largest sphere he can make of the remaining wooden block?

Submit your answer as $\lfloor 10000 r \rfloor$ .

Try also Woodworks

The answer is 7495.

1 solution

Michael Mendrin
Jan 25, 2016

First figure out the volume $V$ of the original tetrahedron, and the area of its four faces. Let $r$ be the radius of the sphere that can be carved from the truncated tetrahedron. We solve for $r$

$V=\dfrac { 1 }{ 3 } \left( \Delta ABC\left( r+1.0 \right) +\Delta ABD(r+0.25)+\Delta ACD(r+0.75)+\Delta BCD(r+0.5) \right)$

which works out to $r=0.749513...$

The method for finding the volume can be found here

It is quite brilliant. I was looking for something like this.

Maria Kozlowska - 5 years, 4 months ago

Well, once again, you made me think twice! First time, I used a flawed approach and got the wrong answer. I took a shower, and then figured out a better way to do it.

Michael Mendrin - 5 years, 4 months ago

I solved it working out scaling ratio for each height but I knew there is a simpler way to do it.

Maria Kozlowska - 5 years, 4 months ago

Cool method. Considering the tetrahedra formed in the original tetrahedron by the incentre of the new tetrahedron really cuts the work down to next to nothing!

I did it the hard way (but I let my computer do the calculations). I determined the coordinates of the points $A,B,C,D$ in some coordinate system, then the equations of the four planes of the tetrahedron, and so the equations of the four planes of the new tetrahedron. I then solved four sets of simultaneous equations to determine the coordinates of the vertices of the new tetrahedron, from which point calculating the volume and surface area, hence inradius, of the new tetrahedron was easy.

Mark Hennings - 5 years, 4 months ago

Woodworking Again

The answer is 7495.

1 solution

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