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Optimization - Rowing and Running

Optimization - Rowing and Running
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2015-04-02 23:31
FunctionCalculusDerivativeOptimizationSlider Animation
① This example visualizes a classic optimization problem: a person starts from a point on shore, rows across water, then runs along the shore to a destination, and we seek the optimal transition point that minimizes total travel time. ② Core mathematical concepts include function modeling, derivative-based extremum finding, optimization, and piecewise time calculation. ③ Construction highlights: using GeoGebra commands such as Point, Intersect, Segment, Min, and CurveCartesian, it builds the rowing path, running path, total time function curve, and uses sliders to adjust rowing speed, running speed, and target position, dynamically showing the optimal path and corresponding time. ④ Suitable for high school to college calculus or mathematical modeling courses, to intuitively grasp the geometric meaning and numerical solution of optimization problems.

GeoGebra Commands & Parameters

  1. Point(xAxis) → B
  2. Point(yAxis) → A
  3. Intersect(yAxis, xAxis) → O
  4. Segment(O, B) → a
  5. Point(a) → P
  6. Segment(P, B) → runpath
  7. Segment(A, P) → rowpath
  8. Min((2 * rowspeed / sqrt(runspeed^(2) - rowspeed^(2))), 6) → xoptimal
  9. CurveCartesian(xRegularPath(t), yRegularPath(t), t, 0, T) → b
  10. CurveCartesian(xOptimalPath(t), yOptimalPath(t), t, 0, T) → b_2
  11. Segment(A, Popt) → rowpathoptimal
  12. Segment(Popt, B) → runpathoptimal

AI Drawing Prompt Reference

Draw a coordinate plane with the horizontal axis as the shore and the vertical axis as the water starting point. Place a target point on the horizontal axis and a starting point on the vertical axis, with the origin at their intersection. From the start, row across water to a point on the shore, then run along the shore to the target. Use sliders to adjust rowing speed, running speed, and target position. Plot the total travel time as a function of the transition point, mark the optimal transition point that minimizes time, and show the corresponding optimal rowing and running paths.
The prompt above was compiled from the original drawing commands and is for reference only.