In Georgian (ქართული): https://drive.google.com/file/d/0B_4P3RNorJzTN3h0Z3cwaG04SXc

Real life math problem and solution

Crossing The Road As A Pedestrian:

The car has n times more speed than the pedestrian. What is the best α angle from the perpendicular to maximize the chance of crossing the road in time before the car approaches?

Restrictions:

1) n > 1

2) 0 ≤ α < 90°

3) The width of the road is unknown

4) The distance between the car and the pedestrian is unknown

Solution:

In order to get the best angle with maximum probability of crossing the road before the car approaches ---- when the pedestrian crosses the road, the distance between the car and him/her must be maximum:

First, define the variables:

Speed of the pedestrian — V

Speed of the car — n⋅V

Horizontal speed of the pedestrian — V_x

Vertical speed of the pedestrian — V_y

Horizontal distance between the first and the last point of the pedestrian — a (not α)

Horizontal distance between the car and the pedestrian — b

Width of the road — d

Time spent to cross the road — t

b affects the maximum, but it does not affect the best angle to get maximum, so we can remove it.

d affects the maximum, but it does not affect the best angle to get maximum, so we can remove it.

In order to find the extremum of a function, we must equalize its derivative to zero:

cos²(α) cannot be zero, so α cannot be 90°, and therefore:

And finally, α = arcsin(1/n) because we already knew that 0 ≤ α < 90° and n > 1.

Answer:

α = arcsin(1/n)

n (n > 1)	α (0 ≤ α < 90°)
1.1	≈65.380023°
1.5	≈41.810315°
2	=30°
3	≈19.471221°
4	≈14.477512°
5	≈11.536959°
10	≈5.73917°
15	≈3.822554°
20	≈2.865984°
50	≈1.145992°
100	≈0.572967°
1000	≈0.057296°

Calculator used

Geogebra plot:

y = b + d ⋅ tan(α) - n⋅V⋅d/(V ⋅ cos(α))

and

α = arcsin(1/n)

https://www.geogebra.org/m/YV7g2YKA