Ken's Puzzle of the Week

Escorting Shapes

  1. A police motorcyclist is escorting a 20 mile line of VIP vehicles. He drives at constant speed from his starting place at the back to the front of the line, and immediately returns with the same speed to the back of the line. He arrives at the back just as the line has moved 20 miles. How many miles did the policeman travel?
  2. A destroyer is escorting a convoy of ships 20 miles long and 20 miles wide. The convoy is in the shape of a square. Starting from the rear left corner of the square, it steams to the front; then steams across the front of the square to the front right; then back down the right side and finally across the rear back to the left rear corner, arriving there just as the convoy has moved 20 miles. How many miles did the destroyer travel?
  3. Same problem as previous, except the convoy is in the shape of an equilateral triangle, side length = 20. (There are three ways to consider this problem: point forward, point rear, and point side.)
  4. Same problem as previous, except the convoy is in the shape of a circle.

 

 

1.        The police motorcycle will approach the front of the line with a speed of s-20 (relative to the line itself), and return to the rear with a relative speed of s+20.  His times for each 20 mile portion are 20/(s-20) and 20/(s+20), respectively.

 

The equation for the time required is then

The solution is

 

2.        The ship has two new directions: along the front and along the back.  If it has speed=s its relative speed along the front will be and it will be the same along the back.

 

The equation to be solved is

The approximate solution is 83.62

 

3.        The relative speed along an edge at angle q to the direction of travel is travelling with the edge, and against it.

 

Case 1: Pointing up (same as pointing down), the angles are 30 (with), 30(against), 90 (perpendicular as in part 2)

The equation is

The approximate solution is 64.721139

 

Case 2: Pointing Left (solution, suprisingly, is same as pointing right), the angles are 60 (with), 60 (with), 180 (against)

The equation is

The approximate solution is 64.72136

 

4.        I decided the circle should have a radius=10.  My above formula give the instantaneous relative speed at angle q.  The solution is the value of s which satisfies the equation
The approximate solution is 67.3614905