# calculus

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A lighthouse is built on an exposed reef, 5.5 miles off-shore. The shoreline is perfectly straight, and a town is located 6.5 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 1.2 miles per hour, and run at 3.4 miles per hour. How far downshore towards the town should he land, if he wants to get to the town as quickly as possible?

• calculus -

56

• calculus -

distance of boat rowing = sqrt(5.5^2 + x^2)
distance of running = 6.5 - x

total travel time

t = sqrt(5.5^2 + x^2)/1.2 + (6.5 - x)/3.4
dt/dx = x/(1.2*sqrt(5.5^2 + x^2) - 1/3.4

d^2t/dx^2 = +ve at any x
x is at its minimum when dt/dx=0

x/(1.2*sqrt(5.5^2 + x^2) = 1/3.4
x = 6/sqrt(253) * 5.5 = 2.075 miles

t = 4.899 + 1.302 = 6.201 hours

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