When is the distance between two walkers at its maximum?

[ufotofu906]

Homework Statement

Two walkers start at the same time from the same place and travel in the same direction with velocities given by
A(t) = 1 - e^-t miles per minute and B(t) = 0.2(e^t - 1) miles per minute and t > 0. They travel until they have the same velocity.

At what time is the distance between them the greatest? Explain your reasoning.

Homework Equations

The Attempt at a Solution

$$\int$$ (1 - e^-t) - $$\int$$ 0.2(e^t - 1)

^^^both integrals are from 0 to t.

That was the expression I came up with to find the distance between the walkers but I don't know what to do with it to find the greatest distance. I tried finding the maximum and minimum of the graphs but that didn't really do anything for me. So basically, I'm pretty much clueless right now. Any ideas?

 

[Gavins]

You're pretty close. Find the distance between the two walkers which is the integral you have. Evaluate that for some t. Then you can differentiate and set to zero to find the maximum.

 

[ufotofu906]

Alright, so I'm not sure if I'm reading what you wrote correctly but basically i had

(1 - e^-t) - 0.2(e^t - 1) = 0

to find the critical points.

I get t = 0 and t = 1.60944.

So I'm guessing it's at t = 1.60944?

 

[Mentallic]

Yes, but you can solve it algebraically. It is essentially a quadratic in e^t once you multiply through by e^t.

Can you reason why the greatest distance between them is when their velocities are equal at the end?

 

[ufotofu906]

I used the first derivative test and at t = 1.5 got a positive result and at t = 1.7 got a negative one so that means at t = 1.60944 there is a maximum on the original function. Do you think that's enough to show why the greatest distance between the two walkers is at t = 1.60944?

 

[Mentallic]

No, showing it is a maximum by using the derivative test or taking the second derivative is common protocol. When it asks to explain your reasoning, it means you need to think of this question as a real life problem (which is why they used walkers in the question) and answer it using common sense with some reference to the math.

Also, can you find the exact answer rather than a numerical solution?

 

[ufotofu906]

Uhh, is it because they have the greatest distance when both their velocities are the same because Person A had traveled at a higher velocity than Person B and thus traveled a greater distance that added up the most at the end of their walk?