Calculus, Optimization: Ship A is 60 miles south of ship B and is sailing North at a rate of 21 mph. If ship B is sailing west at a rate of 22 mph, in how many hours will the distance between the two ships be minimized?

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Answer 1 · 2017-02-10T06:33:29

#t =1260/925 = 1.362# #hr #

Primary equation: #c^2=a^2 + b^2#
Constraint Equations: #a = 60-21t, b = 22t#

Substitute constraint equations into the primary equation:
#c^2 = (60-21t)^2 + (22t)^2#
#c^2 = (60-21t)^2 + 484t^2#

Find First derivative:

#2c c'= 2(60-21t)(-21) +2(484)t#
#c c'= (60-21t)(-21) +484t#
#c c'= -1260+441t +484t#
#c c'= -1260+925t#

#c'= (-1260+925t)/c# = #c'= (-1260+925t)/sqrt((60-21t)^2+(22t)^2)#

Find critical number: #c'=0#
Look at numerator:

#-1260 + 925t = 0#
#1260 = 925t#
#t = 1260/925 hr = 1.362 # #hr#

#Let # #a = 0, 60 = 21t; # # tmax = 60/21 = 2.857 # #hr#

Do a 1st derivative test to see if #t# is a minimum

#Intervals: (0, 1.362), (1.362, 2.857)#
#Let # #t=1 # #and t = 2#
#c'(1)=-7.48 < 0, # #c'(2) =12.41 >0#

Therefore, #t=1.362 # #hr # is a relative minimum