The Sine River is located between y = sin(x) and y = 1 + sin(x) (x in radians, not degrees).
Your starting point is on the shore, at location (0,0) and your destination is location (5,1).
Your running speed is 1 unit per minute.
Your swimming speed is 1/2 units per minute.
Your plan is to divide the trip into three legs (run, then swim, then run) each leg being a straight line. Assuming there is no current in the river (and no wind), what is the minimum travel time to the destination, and by what pathway?
First, the time can be given as a function of two variables. I chose a = the x-coordinate of entering the water, b = the x-coordinate of leaving the water.
This graph shows the time function as eq. 8
https://www.desmos.com/calculator/sbxmsw81ba
It's currently minimized, but you can use the sliders change the trajectory.
I then tried graphing my f(a,b) (x=a, y=b) in Desmos 3D, but it's hard to pin down more decimal places of accuracy.
https://www.desmos.com/3d/wxgjhdx38s
Finally, I tried calculus, to find the critical points of a function of two variables, you can take partial derivatives in terms of x, and y, set each equal to zero and solve the resulting system.
Solving this system by hand proved to be too much, but WolframAlpha gave a high accuracy solution, but no closed form. (I could have zoomed in on the Desmos graph as well)
a=2.7547234707
b=3.40923551154
This graph of the two partial derivatives = 0 is very cool looking.
https://www.desmos.com/calculator/dmd9slennz
I tried using it's symmety to solve it, but no luck.
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Posted by Jer
on 2024-06-14 13:57:26 |
Partial analytic solution w/graphs
