The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter.
Let the place that they touch be x meters from the 300 m wall
Then the distance from the 500 m wall will be 1200 - x
Then....the total distance, D, can be expressed as
D = sqrt [ x^2 + 300^2 ] + sqrt [ (1200 - x)^2 + 500^2 ]
This can be solved with Calculus, but it's a little messy
A graph seems better :
The graph shows that the minimum distance is achieved when x = 450m
And the minimum distance ≈ 1442 m
By the way......we can show that this is true if
arctan [ 300 / 450] = arctan [ 500 / 750 ] ......and this is true !!!!