'With “Rogue One” out, perhaps you’ve been inspired to build your own Death Star.2 Let’s say the fictional Death Star, a spherical galactic superweapon that can destroy planets with its laser, has a diameter of about 150 kilometers, with a two-meter-wide exhaust port at the bottom. Its design is incredibly energy efficient, because nearly all of the energy generated by the reactor is converted into the superlaser’s light, and only a tiny fraction is released as exhaust heat.
After some research, you learn that it is possible to build a Death Star replica 50 meters in diameter, with a one-meter-wide exhaust port. But you don’t want to stop at 50 meters: you want to take the blueprint and scale it as large as possible (e.g., to hundreds of kilometers). There is a significant hitch, though: Power output scales with the volume, while surface area scales only with the radius squared. If you tried to scale the 50-meter replica up, using the same rate of exhaust surface area per unit power, at what radius would the entire surface of the sphere be required for exhaust? (This would become a theoretical upper limit on the radius, and therefore on power output.)
Thought this we would be a good place to put it. Unless I'm wrong, I think this is just a graphing problem, with 196.35 km as the solution. I'm not sure why they posted one so easy, though.
