Saturday Teaser Curio #66: Rolling in Gifts

Did you know you can buy gift subscriptions for as little as $24.99 for your friends and family this holiday? Not only will they love the gift of learning, but you won't have to deal with shipping anything.

Strange shipping rules are the law of the land here in Frustratopia. Congress dictated that all packages must be placed in a rectangular box, then that box is placed in a spherical ball so it can be rolled to its destination. Ron wants to send his honey as much organic vicuña wool as possible this holiday and is trying to figure how much he can send. What is the most volume of wool Ron can send in a rectangular box inside a shipping sphere that's 2 meters across?

Need a hint? If you remember your Pythagorean Theorem, the square of the length of the diagonal of a rectangular box is the sum of the squares of the dimensions. In other words:

D² = L² + W² + H²

Whew! A Curious gift subscription was probably the way to go.

Ready for the answer?

So the most important thing here is to realize that the largest rectangular box you can fit in a sphere is a cube. It makes sense, but if you want, you can prove it using calculus and differential equations. That's WAY out of the scope of the answers here, but the nerd in me wants you to know the proof is super cool (and super nerdy)!

Anyway, given that we're talking about a cube (all three dimensions are the same), we know the diameter of the sphere is the same as the diagonal of the cube. So, using the equation given in the question...

• D² = L² + W² + H²
• D² = 3L²
• 2² = 3L²
• 4 = 3L²
• L = (4/3)^½

And the volume of that cube is the cube of that length: L³ ~= 1.5396... cubic meters.

That's a LOT of wool: over 400 gallons of it!