A sailboat is sailing on a lake with a constant speed of 10 miles per hour. At noon, the sailboat is located at the point (0,0) on a Cartesian coordinate grid, and it is heading due east. At the same time, a motorboat is located at the point (0,10) on the grid, and it is traveling in a circle with a radius of 10 miles at a constant speed of 20 miles per hour.
Topic: Time Rates: Trigonometric functions
A sailboat is sailing on a lake with a constant speed of 10 miles per hour. At noon, the sailboat is located at the point (0,0) on a Cartesian coordinate grid, and it is heading due east. At the same time, a motorboat is located at the point (0,10) on the grid, and it is traveling in a circle with a radius of 10 miles at a constant speed of 20 miles per hour.
If we let t represent the time in hours since noon, then we can use trigonometric functions to represent the positions of the two boats at time t. The position of the sailboat at time t is given by the point (10t,0), and the position of the motorboat at time t is given by the point (10cos(t), 10sin(t)).
When will the two boats be closest to each other, and what is the minimum distance between the boats?
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