# Calculus

A lighthouse is built on an exposed reef, 5 miles off-shore. The shoreline is perfectly straight, and a town is located 9 miles downshore from the point on the shoreline nearest the lighthouse.
The lighthouse keeper needs to go from the lighthouse to the town to get fresh supplies. He can row a boat at 1.8 miles per hour, and run at 3.6 miles per hour. How far downshore towards the town should he land, if he wants to get to the town as quickly as possible?

Distance downshore to the landing point =
sqrt(25/3) miles.

Time it takes to get from the lighthouse to the town = 4.906 hours.

After several such trips, the lighthouse keeper buys a motor boat, which can go 6 miles per hour.
How far downshore should he land if he uses his motorboat, instead of rowing?

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1. Just substitute 6 for 1.8 in your time function. You will find that the shortest time is to boat all the way.

A little analysis will show you that if the boat is faster than the running, the boat is the best way. If the ratio of the boat:run speed is k, then the distance downshore for fastest time is x=5k/√(1-k^2)

In the original problem, k = 1/2, so x = 5/√3 as you already found.

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