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?

Question

6089 views around the world

You can reuse this answer
Creative Commons License

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