A strong swimmer is one mile off shore at point P. Assume that he swims at the rate of 3 mile per hour and walk at the rate of 5 miles per hour. If he swims in a straight line from P to R and then walks from R to Q, how far should R be from Q in order that he arrives at Q in the shortest possible time?
2 answers
I have no idea where Q is
If he lands x miles from the nearest point on the shore to P (call it O), and O is m miles from Q, then the distance traveled is
z = √(x^2+1) + m-x
The time taken is thus
t = 1/3 √(x^2+1) + (m-x)/5
You want to minimize t, so
dt/dx = x/3√(x^2+1) - 1/5
dt/dx = 0 when x = 3/4
So, time is minimum when x = 3/4 mile from O.
It is interesting that it doesn't really matter how far O is from Q. What matters is the distance from O to R, compared to the distance from O to P. That and the ratio of swimming/walking speeds.
z = √(x^2+1) + m-x
The time taken is thus
t = 1/3 √(x^2+1) + (m-x)/5
You want to minimize t, so
dt/dx = x/3√(x^2+1) - 1/5
dt/dx = 0 when x = 3/4
So, time is minimum when x = 3/4 mile from O.
It is interesting that it doesn't really matter how far O is from Q. What matters is the distance from O to R, compared to the distance from O to P. That and the ratio of swimming/walking speeds.
2 answers