OPTIMIZATION PROBLEM : BOAT?

let t = time after 2 pm

put dock @ origin of xy-plane.

position of southward-moving boat @ time t

y = -20t

eastward-moving boat reaches dock @ 3 pm after travelling 10 km/hr. @ 2 pm located @ position x = - 10 on x-axis. it's position @ time t

x = 10t - 10

distance between them minimum when square of distance minimum (this saves taking derivatives messy square roots). want minimize x^2 + y^2, or

f(t) = (10t - 10)^2 + (-20t)^2 = 100(t-1)^2 + 400t^2

f '(t) = 200(t-1) + 800t = 1000t - 200

1000t - 200 = 0 when t = .2

.2hrs = .2(60) min = 12 minutes after 2 pm

@ 2:12 pm closest.

a boat leaves dock @ 2:00 p.m. , travels due south @ speed of 20 km/h. boat has been heading due east @ 10 km/h , reaches same dock @ 3:00 p.m. how many minutes past 2:00 p.m. boats closest together?

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