Question 545126
At noon, the cruise ship Celebration is 60 miles due south of the cruise ship Inspiration and is sailing north at a rate of 30 mph.
 If the inspiration is sailing west at a rate of 20 mph, find the time at which the distance d between the ships is a minimum.
:
let t = the travel time in hrs of both ships
then
30t = travel dist of the Celebration (sailing north toward the ref point)
20t = travel dist of the Inspiration (sailing west from the ref point)
:
The two ships course form a right triangle from the ref point
The distance between the two ships, is the hypotenuse (d)
d^2 = (60-30t)^2 + (20t)^2
:
d^2 = 3600 - 1800t - 1800t + 900t^2 + 400t^2
d^2 = 3600 - 3600t + 1300t^2
d = {{{sqrt(1300t^2 - 3600t + 3600)}}}
we can ignore the square root when we find the axis of symmetry
t = {{{-(-3600)/(2*1300)}}}
t = {{{3600/2600}}} = {{{18/13}}}
t = 1.3846 hrs the distance between them will be minimum
:
What is this distance?
Find the distance each ship is from the ref point in 1.3846 hrs
60-(30*1.3846) = 18.46, northbound Celebration
1.3846*20 = 27.7, westbound Inspiration
:
Find the distance between the ships at this time
d = {{{sqrt(18.46^2 + 27.7^2)}}}
d = 33.3 mi apart at this time