5&6.1 Optimal Path (#differentiation)

You are a rover pilot on the crew of the initial exploration team sent to Kepler 22b,

the first extrasolar planet discovered within the habitable zone of a sun-like star. The

science team recently discovered liquid water on the surface. (Hurrah!) Your rover is

at point A on the shore of a circular lake with radius 4 km collecting samples. The

science team wants to send your rover to a point C diametrically opposite A. The

rover can drive around the circumference of the lake at a rate of 4 km per hour and

fly over the lake at a rate of 3 km per hour.

(a) How long will it take the rover to fly across the lake?

(b) How long will it take the rover to drive around the shore of the lake?

You could also fly at an angle 0 along a chord inside the circular lake, and

complete the rest of the path driving along the circumference of the lake.

(c) Find the length of the chord in terms of 0. How long will it take the drone to

traverse the chord?

(d) Find the length of the remaining shoreline after the cord in terms of 0. How

long will it take the drone to drive along the remaining shoreline?

(e) Combine your answers to the last two parts to get a function for time traversing

the lake in terms of 0.

(f) What is the choice of that would minimize the time the rover takes to get to

point C? What value of 0 would maximize the time she takes to get to point

C?

(g) How will you pilot the drone?