Solved: Related Rates - Find Distance & Time Closest Approach

[Sheneron]

[SOLVED] Related Rates

Homework Statement

Two aircraft are in the vicinity of a control center. Both are at the same altitude. Plane 1 is 36 nautical miles from the center and approaching it at a rate of 410 knots. Plane 2 is 41 nautical miles from the center and approaching it at a rate of 455 knots. (One knot is 1 nautical mile per hour)

A) How close will the planes come to each other?
B) How many minutes before the time of closest approach?

The Attempt at a Solution

I can't figure out how to solve this... I keep getting stuck here is what I have done.

$$P_1(t)^2 + P_2(t)^2 = D(t)^2$$

take the derivative

$$2P_1(t)P_1'(t) + 2P_2(t)P_2'(t) = 2D(t)D'(t)$$

and then the place where the distance would be a minimum is where D'(t) = 0, but I keep getting stuck here because I don't know either of the two P(t)s. Can someone please help me set this problem correctly? Thanks

 

[Sheneron]

I still can't figure this out, and I think I am not going about it properly. If someone could help me set it up I would appreciate it. Thanks

 

[Dick]

You can't do that at all unless you know something about the angle between the two planes approach paths. I'm guessing since you using Pythagoras that they are coming in at right angles? If so then just try to write P1 and P2 as explicit functions of t. They are linear functions (since velocity is constant). Try and start with P1(t). P1(0)=36mi. P1'(t)=410mi/hr, right? What does P1(t) look like?

 

[Sheneron]

P1(t) = -410t + 36

 

[Sheneron]

Now that I have two explicit functions of t, assuming that I did so correctly, what do I do next?

 

[Dick]

P1(t) = -410t + 36

Great! Now do P2(t) and put them into your equation.

 

[Sheneron]

ahhh yes i just saw it I think I have it now!

Thanks... I got it