Practice problem
Area and Perimeter of Similar Figures
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The Death Star, a galactic superweapon that can destroy planets with its laser, has a diameter of about 150 kilometers, with a 2-meter-wide exhaust port at the bottom. This 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 dissipated as exhaust heat.
To compare this with today's technology, suppose that we can build a version 50 meters in diameter, with a 1-meter-wide exhaust port. If we used this level of technology to power a 150-kilometer-diameter version, where power output grows proportionally with the volume, what fraction of the surface area will be taken up by the exhaust port?