All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
The toll of two cities (Posted on 2017-02-20) Difficulty: 3 of 5
The two cities, A(0,a) and B(b,-c), are separated by a river (river's banks are defined by y=0 and y=a/10). A bridge, about to be erected must be perpendicular to river's banks (i.e. line x=d).

Evaluate d to minimize the travel distance between A and B.

See The Solution Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution Comment 1 of 1
There is a translation trick to find the slope easily: translate the river until one of its banks coincides with city A.  Then find the slope of the line from the other end of the bridge to city B.  Both sloped roads in the original configuration will share that slope.

After translation other end of the bridge is at (0, 9a/10).  The slope from this point to B at (b,-c) is -(9a/10 + c)/b.

Back to the original configuration.  The line of the road starting at city A is then y - a =  -(9a/10 + c)/b * (x - 0), which simplifies to y =  -(9a/10 + c)/b * x + a.  
The line of the road starting at city B is then y - -c = -(9a/10 + c)/b * (x - b), which simplifies to y = -(9a/10 + c)/b * (x -b) - c.

Evaluate the first line at y=a/10 and the second line at y=0.  These should give the same answer for x, the expression for d.

The first line becomes a/10 =  -(9a/10 + c)/b * x + a, which simplifies to x=9ab/(9a+10c).
The second line becomes 0 = -(9a/10 + c)/b * (x -b) - c, which also simplifies to x=9ab/(9a+10c).

Then the answer is d = 9ab/(9a+10c).

  Posted by Brian Smith on 2017-02-20 10:08:15
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information