A river flows at the speed of v from west to east. How should a man who is at a point A of the southern bank of the river, which is d meters away from the river, and wants to reach a point B on the northern bank of the river, which is 2d meters away from the river, choose his path in order to reach the destination at the shortest time? Assume the width of the river to be d, the man to run at a speed of 3v and swim at the speed of v/3 in still water, and the points A and B on the same meridian.
So, finish the diagram. If the man enters the water at point C, x meters upstream from A, and lands at point D, y meters downstream from B, then we have distances of
land: (d^2+x^2)
water: √(x+y)^2+d^2)
land: √(y^2+(2d)^2)
The time it takes to cross the river is shortest when he swims directly north, taking d/(v/3) = 3d/v seconds. In that time, he drifts downstream at speed v, making y = 3d meters. So, now we see that the distances are
√(d^2+x^2) + √(d^2+16x^2) + √(9x^2+4d^2)
Now, divide each distance by its corresponding speed, and we have
1/(3v) (√(d^2+x^2) + 9√(d^2+16x^2) + √(9x^2+4d^2))
That's all very messy, but the v does not affect anything at this point. So, let us express x in terms of d. That is, consider the ratio x/d, rather than an absolute distance. If we let u = x/d, then we have the time
t = d/(3v) (√(1+u^2) + 9√(1+16u^2) + √(4+9u^2))
To minimize the time, we must have dt/du=0. It's a bit messy, but we find that dt/du=0 when u=0. That is, the man runs directly from A to the river, swims right across, then runs from where he lands on to B.
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Let's say he swims from A directly north, being taken downstream d/(v/3)*v meters to point C. Then his distances are
d on land from A to river
d north in water
√((3d)^2 + (2d)^2) on land to B
time: d/3v + d/(v/3) + d√13/3v
= d/v (1/3 + 3 + √13/3)
= dv/3 (10+√13)
To find the path that will minimize the time it takes for the man to reach point B on the northern bank of the river from point A on the southern bank, we need to consider the man's speed both on land and in the water.
Let's break down the journey into two parts: crossing the river and running on land.
1. Crossing the River:
Since the river flows from west to east, the man needs to swim across it. The speed of the river is v, and the man swims at a speed of v/3 in still water. To minimize the time spent swimming, the man should swim directly across the river, along the shortest path.
2. Running on Land:
After crossing the river, the man needs to run from the riverbank to point B on the northern bank. The total distance the man needs to cover on land is 2d meters.
The man's running speed is 3v, which is faster than his swimming speed. Therefore, it is more efficient to spend less time swimming and more time running. So the optimal strategy is to swim directly across the river to the point where the man would cross the river if it were still, and then run directly to point B.
To calculate the total time taken, we need to consider both the swimming and running times.
Swimming Time:
The time taken to cross the river can be calculated using the formula: time = distance / speed. In this case, the distance to be crossed is d meters, and the swimming speed is v/3. So, the swimming time (t_swim) is:
t_swim = d / (v/3)
= 3d / v
Running Time:
The time taken to run on land is given by the formula: time = distance / speed. In this case, the distance to be covered on land is 2d meters, and the running speed is 3v. So, the running time (t_run) is:
t_run = (2d) / (3v)
= 2d / (3v)
Total Time:
To find the total time, we add the swimming time and the running time together:
t_total = t_swim + t_run
= 3d/v + 2d/(3v)
= (9d + 2d)/(3v)
= 11d / (3v)
Therefore, the path that will allow the man to reach point B on the northern bank of the river in the shortest time is by swimming directly across the river to the point where he would have crossed if the river were still, and then running directly to point B. The total time taken will be 11d / (3v).