Alexander: Straight road and straight railroad intersect at angle 60 degrees?
A city is located 10 miles away from intersection somewhere between the roads.
City council decided to bulid bus station A, and train station B, and connect them to the city by a local shuttle.
The shuttle route begins at city hall plaza O, runs to bus station A, then to train station B, then back to the plaza O.
What is minimal length of shuttle route?
Answers and Views:
Answer by Arin
Your triangle that forms the minimal distance is a equalateral triangle. All side will be equal and it will cut the 10 miles in half. so your triangles height will be 5 miles. sides lengths will be 2.5*sqrt(3) or 4.33 miles which mean it travels 7.5*sqrt(3) or 12.99 miles
only happening if city hall plaza O have equal distance from road and railroad. and the route makes equilateral triangle.
height of triangle = 10/2 = 5
side = 5*(2/√3) = 10/√3
minimal shuttle route
= 3 sides
= 10√3
edit
zo maar is good. i gave a thumb.
The minimal length of the route is 10*sqrt(3) miles, independently on the position of the city.
The shortest route should immitate the trajectory of light. The shuttle goes to the road, “reflects” (angle of incidence is equal to the angle of reflection), goes to the railroad, “reflects”, and goes back to the city. These reflections are convenient to visualize by making “mirror images” of the original angle formed by the road and the railroad. If they form an angle with lines ED and EF, and the shuttle goes to the line ED, then we reflect the angle DEF with respect to the line ED and build its “mirror” DEG image. The mirror image of the trajectory of the shuttle will be a straight line, that goes from the sector DEF to the “mirror” sector DEG. Similarly, we build another “mirror” sector, by reflecting DEG with respect to EG.
The resulting trajectory of the shuttle, visualized in this way, will be a straight line, passing through three sectors. This straight line is the third side of an isosceles triangle, whose two sides are equal to 10 miles and they form the angle of 120 degrees. This gives the answer.
Leave a Reply