Solve the problem below and explain fully your reasoning Giv

Solve the problem below and explain fully your reasoning:

Given :

A) A man walking across a train bridge is 0.65 of the way across the bridge when he hears a train coming.

B) The speed of the train (vt) = 60 mph.

C) The man can run at top speed (constant) in either direction and just get off of the bridge before the train runs over him

Problem: How fast can the man run?

Solution

Let the length of the bridge be miles. Then the man has covered 0.65 miles acroos the bridge before he hears a train coming. Let the distance of the train from the end of the bridge nearer to the man be z miles when the man hears the train coming. Since the speed of the sound is 768 mph, the train would have travelled a further distance of ( z + 0.35x)/768 * ( 60) = 5/64(z + 0.35x)miles before the man starts running. At this tme, the distance between the man and the train is (z + 0.35x) - 5/64( z + 0.35x) = 59/64 ( z + 0.35x). Let the speed of the man be y mph in either direction. If the man runs towards the train, he will cover the distance of 0.35x in (1/y)(0.35x) hours. In this time, if the man just gets off the bridge before the train enters the bridge, the train will cover the distance of [59/64( z + 0.35x) - 0.35x] = [(59/64)z - 5/64(0.35x)] = 1/64[ 59z - 1.75x] miles . The train will cover this distance in (1/64*60)[ 59z - 1.75x] hours.   Therefore, (1/y)(0.35x) = (1/64*60)[ 59z - 1.75x] ....(1) If the man will run towards that end of the bridge where he started from and the train comes behind him, then the man covers 0.65x miles in 0.65x/y hours. The train is travelling at a speed of 60 miles per hour. so that the train will cover x + 1/64[ 59z - 1.75x]   miles in [x + 1/64{ 59z -1.75x}/60hours. Therefore,   0.65x/y = [x + 1/64{ 59z- 1.75x} /60...(2) ot make any mention of the distance of the train from the man when he hears it coming, we have presumed that the train is at the beginning of the bridge. On simplification, the equations 1 and 2 become 2496x = 59yz + 62.25xy...(3) and 1344x = 59yz - 1.75xy...(4). On subtracting the 4th equation from the 3rd equation, we get 1344x = 64xy so that y = 1344/64 = 18mph. The answer is the the man can run at a speed of 18 mph.