Limits, differentiation, related rates, integration, trig integrals, etc.

This is a problem in an old calculus text book from the 1970's.
The text gives the answer to the problem but not the way in which it was calculated.
Two ships leave the same point going in different directions and leaving at different times.
Ship A leaves at 10 am going north at 12 mph
Ship B leaves at 10:30 am going west at 16 mph
Find the Time when the rate of change of the distance between the ships equals 18.86 mph
The distance between the ships (S) is the hypotenuse ...we want to find the time when dS/dt =18.86

Here's what I did to try and solve it:

Put everything in terms of t
Distance ship A goes is 12 t + 6
Distance ship B goes is 16 t
dA/dt=12 mph. dB/dt=16 mph and dS/dt=18.86
When A= 12t, B=16t-8
When A=12t+6, B =16 t
Getting S in terms of t Similar triangles: 12t + 6/S = 6/10 (10 is from Pythagoras with 6 vertical and 8 horizontal)
(120t + 6)10 = 6S
Substituting into S dS/ dt = A dA/dt + B dB/dt
(20t + 10) 18.86 = (12t + 6)12 + 16t* 16
377.2t+ 188.6 =144t+72+256t
377.2t + 188.6 = 400t + 72
22.8t=116.6
t=5.11
And this is the wrong answer. Can anyone tell me how this problem is Solved?
Last edited by raymo on Sun Oct 20, 2013 3:31 pm, edited 3 times in total.
raymo

Posts: 1
Joined: Mon Oct 14, 2013 11:35 pm

### Re: Related rate right triangle find the time.

raymo wrote:Ship A and ship B leave the same point. A leaves at 10am going 12 mph north. B leaves at 10:30 am going 16 mph west. At what time is the rate of separation between the two ships 18.86 mph?
This problem was in an old calculus book from the 1970's. The other part of the problem was a standard do-able problem,
What is the rate of separation at 11:30 am? Implicitly Differentiate Pythagoras , plug in calculated distances ( ship A 1 and 1/2 hours times 12 mph, ship B 1 hour times 16 mph. And the answer is 19.6 mpr rate of separation.
The first part given the rate of separation and asking for the time it happens has proven to be tricky.

Which part is the "first part" that you're having trouble with?

raymo wrote:Every approach I try has failed.... Everything I try ends up as a convoluted mess and when I apply the quadratic formula I get an impossible equation....

Please write back with a sample approach, so we can see what you're doing. Maybe you're almost there!

maggiemagnet

Posts: 289
Joined: Mon Dec 08, 2008 12:32 am