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?
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.
To determine the shortest possible time for the swimmer to arrive at point Q, we need to find the location of point R that minimizes the total travel time.
Let's analyze the situation step by step:
1. The swimmer is currently at point P, one mile offshore.
2. The swimmer swims at a rate of 3 miles per hour. So, it will take him (1 mile / 3 miles per hour) = 1/3 hour to reach point R from P.
Now, we need to find the optimal distance between R and Q to minimize the walking time.
3. The swimmer walks at a rate of 5 miles per hour. So, the walking time can be calculated as the distance between R and Q divided by the walking rate, which is (Distance between R and Q / 5 miles per hour).
We can express the total travel time as the sum of the swimming time and the walking time: T = (1/3) + (Distance between R and Q / 5).
To find the location that minimizes the total travel time, we take the derivative of T with respect to the distance between R and Q, and set it equal to zero.
dT/d(Distance between R and Q) = 0
Differentiating the expression for T:
d((1/3) + (x / 5))/dx = 0, where x represents the distance between R and Q.
Simplifying the equation:
1/5 = 0
Since the derivative is equal to zero, this means that the total travel time T is at its minimum when the distance between R and Q is zero. Therefore, R and Q should be at the same location for the swimmer to arrive at Q in the shortest possible time.
In other words, the swimmer should swim directly from P to Q without making a detour to point R.