Thanks to all who submitted solutions to this week’s puzzle. You are all doomed to be servants of Frunobulax in one way or another, but Zachary Kelton gets an extra turkey leg and can be thankful for a $50 gift code to ThinkGeek! Check the solution and the coupon code that will get you $10 off after the jump.
Special thanks to ThinkGeek for providing our prizes!
It’s not easy being a genius. All sorts of nefarious characters think they can just have use of your brilliance for whatever evil scheme crosses their mind. So it was that you found yourself abducted in the wee hours of the morning and held captive in the evil but very bizarre lair of some character who calls himself Frunobulax.
Blah, blah, he wants you to build a time portal so his henchmen can go anywhere and work their evil deeds, blah, blah, your life is on the line…the usual. And this being the week of Thanksgiving! Luckily you’ve escaped and are making a hasty exit through some sort of utility access-way. Your way is suddenly blocked except for a single perfectly cylindrical tunnel that extends through an empty sphere end-to-end. The sphere is suspended in a vertical conduit with other spheres like it above and below.
The warning pictograms on the wall indicate that to pass through the tunnel, the sphere must be filled with liquid or else it will not support your weight and come crashing down in the seemingly endless drop. The flow of liquid is controlled by a panel on the wall. More warnings reveal that if too much liquid is released, the sphere will come crashing down. If too little, your weight will bring it down as you cross through. The only way to traverse the tunnel is to release the perfect volume of liquid via the panel to fill the sphere. One pictogram shows the length of the tunnel from opening to opening to be
20 meters. Another shows that only one panel input per side of the tunnel is allowed before the panel locks for safety reasons. You hear voices and footsteps and must move fast if you are to escape.
What volume of liquid do you punch in to fill the sphere?
Solution
The radius of the sphere (sans tunnel) is r. The unknown radius of the hole b, the height of the flattened spherical cap c, and half the length of the tunnel hole a. The only one of these items whose value we know is a: 10 meters. See diagram.
If we consider the measurement of the tunnel to be single units, per the Pythagorean theorem, we know that a^2 + b^2 = r^2. Therefore, b= sqrt(r^2-1); and c=r-1.
The volume of the sphere not taken up by the tunnel equals: (the volume of the sphere) – ((the volume of the cylinder) + 2 (the volume of each cap)).
The formula for the volume of the sphere is 4/3 pi r^3. The formula for the volume of the cylinder is (pi * b * 2a). The formula for the volume of the spherical cap is 1/6 pi c (3b^2 + c^2).
Plug in the values so far: (4/3 pi r^3) – ((pi * (sqrt(r^2-1)) * 2) + (1/3 pi (r-1) (3 (sqrt(r^2-1))^2) + (r-1)^2)))
All the r’s cancel, giving you 4/3 pi cubic meters. Plugging in the actual values, you should arrive at 4,189 cubic meters. (NOTE: Depending how far you carried out decimals, you answer may differ, but it’s close enough for the GeekDad puzzleland.)
Once you zip through the tunnel, you reset the panel on the other side. As you’re making your speedy getaway, you hear a violent crash, then another, and another, and smile at your cleverness. Thanksgiving dinner here you come!
Coupon code GEEKPUZZLE will get $10 off of your next ThinkGeek purchase of $50 or more.
