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 θ 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 θ. 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 θ. 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 θ.
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 θ 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 θ. 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 θ. 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 θ.
(f) What is the choice of θ that would minimize the time the rover takes to get to
point C? What value of θ would maximize the time she takes to get to point
C?
(g) How will you pilot the drone?
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