The thermal gradient of Earth's crust is approximately \(25 \mathrm{~K} / \mathrm{km}\) at locations away from tectonic plate boundaries. Suppose you want to use this thermal gradient to run a heat engine that operates on a Carnot cycle at a location where the base of the crust is \(30 \mathrm{~km}\) below Earth's surface and the surface temperature is \(28{ }^{\circ} \mathrm{C}\).

(a) What is the maximum efficiency the engine can achieve at this location?

(b) Because drilling \(30 \mathrm{~km}\) into the crust presents great technological problems, you will settle for an efficiency of 0. 50 . How deep into the crust must your high-temperature reservoir be in order for the engine to have an efficiency of 0. 50 ?