Expert: Paul Klarreich - 12/4/2007

Question
Hi, I'm having a hard time with a related rates problem and i can't seem to find out how to do it anywhere. It's...

A railroad bridge is 20 feet above, and at right angles to, a river. A man in a train traveling 60 mi/hr passes over the center of the bridge at the same instant that a man in a boat traveling 20 mi/hr passes under the center of the bridge. How fast are the two men moving away from each other 10 sec later?

So I converted 60 mi/hr to 88 ft/sec and 20 mi/hr to 29.3 ft/sec, but after that I'm completely lost. Thank you so much!

Answer
Questioner:   Sarah
Category:  Calculus
Private:  No
 
Subject:  related rates!
>> Let's not get too excited, now.

Question:  Hi, I'm having a hard time with a related rates problem and i can't seem to find out how to do it anywhere. It's...

A railroad bridge is 20 feet above, and at right angles to, a river. A man in a train traveling 60 mi/hr passes over the center of the bridge at the same instant that a man in a boat traveling 20 mi/hr passes under the center of the bridge. How fast are the two men moving away from each other 10 sec later?

So I converted 60 mi/hr to 88 ft/sec and 20 mi/hr to 29.3 ft/sec,

>> That's a good start.

but after that I'm completely lost. Thank you so much!
....................................
Hi, Sarah,
I'll try to get you started and you can finish up.


    train
+------------------> 88 ft/sec
|       x
|y
|
V
29.3 f/sec

Let  x = distance traveled by train
    y = """""""""""""""""''' boat
    r = distance between two men

dx/dt = speed of train, GIVEN as 88
dy/dt = speed of boat, GIVEN AS 29.3
dr/dt = rate at which they separate, TO BE FOUND.

Relation:  (Since one is higher by 20 ft.)

x^2 + y^2 + 20^2 = r^2  [Pythag Thm in 3 dimensions.]

Diff:
x dx/dt + y dy/dt = r dr/dt

Values at t = 10

x = 880
y = 293

We need r, so back to the P.T.:

(880)^2 + (293)^2 + 20^2 = r^2

and we start our calculator:

r^2 = 860649

r = 927

Ready to go:
x dx/dt + y dy/dt = r dr/dt

(880) (88) + (293)(29.3) = (927) dr/dt

Ok, you can do the rest, I think.