An Olympic athlete is standing at the edge of one side of a 3.7km wide river and wants to reach the point 7.4km downstream just along the edge of the opposite side. For this entire 7.4km stretch, the river is completely straight. The velocity of the current is negligible. The athlete can swim at 7.4km/hr and run at 14.8km/hr. If she is to swim first and then run second if necessary, how far down the stream should she swim if she is to make it to her destination point in the shortest time? Round your answer to the nearest four decimal places.
If she lands a distance x downstream, then she
swims √(3.7^2+x^2)
runs 7.4-x
So, the time taken is
t(x) = √(3.7^2+x^2)/7.4 + (7.4-x)/14.8
find x where dt/dx=0
I get x = 3.7/√3
All these speeds and distances are creepily multiples of 3.7
To find the shortest time, the athlete needs to minimize the total distance traveled. This means finding the optimal point to start running after swimming a certain distance downstream.
Let's break down the problem:
1. Swim distance: The athlete can swim at a speed of 7.4km/hr, and the river width is 3.7km. Since the current velocity is negligible, the athlete should aim to swim directly across the river perpendicularly to take advantage of the shortest distance. Therefore, the swim distance is equal to the width of the river, which is 3.7km.
2. Run distance: After swimming across the river, the athlete needs to run to the destination point, which is 7.4km downstream. Since the athlete's running speed is faster than her swimming speed, it is advantageous to minimize the running distance.
Now, to find the optimal point to start running, we need to consider the distance the athlete travels swimming and the distance she needs to run.
If we represent the swim distance as x (in km), then the run distance will be the total distance from the starting point to the destination point minus the swim distance, which is (7.4km - x) km.
The total time taken will be the swim time plus the run time. The swim time is equal to the swim distance divided by the swimming speed, which is x/7.4 (in hours). The run time is equal to the run distance divided by the running speed, which is (7.4 - x)/14.8 (in hours).
Therefore, the total time taken is:
Time = swim time + run time = x/7.4 + (7.4 - x)/14.8.
We can simplify the equation:
Time = x/7.4 + (7.4 - x)/14.8
= (2x + 7.4 - x)/14.8
= (7.4 + x)/14.8.
To minimize the time, we can take the first derivative of Time with respect to x and set it equal to zero:
d(Time)/dx = 1/14.8 = 0.
Solving for x, we find:
7.4 + x = 0,
x = -7.4.
Since the value of x cannot be negative, we ignore the negative solution.
Therefore, the athlete should swim a distance of approximately 3.7km downstream to minimize the total time taken to reach the destination point.
Answer: The athlete should swim approximately 3.7km downstream.