Euler Problem 12? Michael and Tom?

Question

Tom was floating down the river on a raft when, 1km lower down, Michael took to the water in a rowing boat. Michael rowed downstream at his fastest pace. Then he turned around and rowed back, arriving at his starting point just as Tom drifted by. If Michael's rowing speed in still water is ten times the speed of the current in the river, what distance had Michael covered before he turned his boat around?

Answer 1

Let x = distance Michael had covered before he turned his boat

around

m = Michael's rowing speed in still water

r = speed of river

According to the problem:

m = 10r

1 km = distance traveled by Tom to reach Michael'

starting point

Therefore,

Speed of Michael downstream = 10r + r = 11r

Speed of Michael upstream = 10r - 4 = 9r

Speed of Tom downstrea = r

Time for Tom to reach starting point of Michael = Time for Michael to return to his starting point

Tt = Tmd + Tmu, where Tt = time of Tom downstream

Tmd = time of Michael downstream

Tmu = time of Michael upstream

But time = distance/speed; hence,

1/r = x/11r + x/9r

x/11r + x/9r = 1/r

x(1/11r + 1/9r) = 1/r

x = (1/r)/[1/(11r) + 1/(9r)]

x = (1/r)/(1/r)[1/11 + 1/9]

x = 1/[1/11 + 1/9]

x = 4.95 km

Hope This helps you, i just completed the euler questions too!

Answer 2

We count the time.

Let the current speed is V, and Michael's speed is 10V

Time when Tom float for 1 km is 1/V = A

Time for Michael rowed back and forth in distance S is:

S/(10V+V) + S/(10V-V) = S(1/11V + 1/9V) = B

A = B

1/V = S(1/11V + 1/9V)

1 = S(20/99)

S = 99/20 km = 4,95 km