A boat leaves a dock at 4:00 PM and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 5:00 PM How many minutes after 4:00 PM were the two boats closest together?

Take the first boat's starting point to be the origin. After some time (in hours), the second boat covers a distance of km.

At 4:00 PM, the second boat is 15 km to the east of the dock. After some time , it covers a distance of km. Its distance to the dock is given by km.

The paths of the boats form a right triangle, in which the hypotenuse (also the distance between the two boats) is

We want to minimize this distance over the time interval .

We have critical points where the first derivative vanishes:

(and at this critical point, we can show the second derivative is positive, indicating a minimum)

The two boats are closest together 0.36 h = 21.6 min after 4:00 PM.

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