Three bank robbers traveling in a gateway car at a constant speed of 70 mi/hr pa

Question

Three bank robbers traveling in a gateway car at a constant speed of 70 mi/hr pass a fruit stand. 4.0s later, a cop car traveling in the same direction at 80 mi/hr passes the same fruit stand. A certain time later, the cop car catches up with the robbers.

Questions: How much time elapses between the robbers passing the fruit stand and the cops catching the robbers? What is the distance between the fruit stand and the place where the cops catch the robbers?      

Using these formulas for constant velocity:

Explanation / Answer

we need to set up two equations, one for the cop, and one for the robbers. The trick is to write the equations in terms of d: the distance traveled by either vehicle, and t: the time either vehicle traveled.

first, the robbers:

we use v = d1/t1. The velocity is 70, and we don't know t or d, so:
70 = d1/t1

Now, the cops,
v=d2/t2
80 = d2/t2.

Now, this seems unsolveable, but lets examine d2. d2 is actually the same as d1, if you read the way the question is worded, the cop has to travel exactly as far as the robbers, with respect to the fruit stand, to catch up to them.

Now, lets examine t2. The time the cop has to travel is 4 seconds LESS than the robbers. In other words, the robbers drove for a time t1, but the cop waited 4 seconds before leaving, so the cop drove for t1 - 4 sec! So now we can solve. I will rewrite the equation for the cop using this new info:

v=d2/t2

80 = d1 / ( t1 - 4 )

Now, if we look at the first equation, lets rewrite it as 70t1 = d1, and then plug this into the second equation:

80 = ( 70t1 ) / ( t1 - 4 )

80t1 - 320 = 70t1

10t1 = 320

t1 = 32 seconds

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