Question

A blind man wanders under a door into an empty room. After running into the walls repeatedly for a while, it stops to think and realizes it must be stuck in a room. So far its strategy has been this: Go in a straight line until you either hit a wall or escape under the door. When you hit a wall rotate a random amount and then start forward again. Repeat this process until escaped. The man moves at a rate of 0.5 cm/sec
A) Direct Escape = give the room in figure image the man starting position (circle) what are its odds of escaping directly on his first move?
B) How long on average will it take the Mab to escape the room through the crack under the door?

Answer 1

From the side where the man is located in the 10' by 12' rectangular room, we have;

A) 0.046

B) The average time to escape in first attempt is approximately 605.02 seconds

Which concept can be used to find the odds and time of escape?

A) From The given diagram, we have;

Width of the door = 10 - 6 - 2 = 2

Therefore, the door is 2 feet wide

Distance, d1, from the man to the side of the door closer to him is given by Pythagorean theorem as follows;

• d1 = √(6² + 7²) = √(85)

From the furthest side of the door to the man, we have;

• d2 = √(7² + 8²) = √(113)

According to the law of cosines, we have;

2² = 85+113 - 2×√(85)×√(113) cos(A)

Where angle A is the angle formed by the lines from the man to the closer and farther sides of the door.

2×√(85)×√(113) cos(A) = 85+113 - 2²

A = arccos((85+113 - 2²)/(2×√(85)×√(113)))

A = arccos (194/(2×√(9605)) ≈ 8.213°

The angle of the possible directions in which the man can turn is 180°.

Therefore;

The probability, P(E), of escaping in the first move is therefore;

• P(E) = 8.213°/180° ≈ 0.046

B) Given that the man escapes in the first move, the shortest and longest distances the man moves are;

d1 = √(85) and d2 = √(113)

Speed of the man, v = 0.5 cm/sec

Therefore by unit conversion function of a graphing calculator;

• v = (25/1524) ft./sec

The times taken are;

t1 = √(85)/(25/1524) ≈ 562.023

t2 = √(113)/(25/1524) ≈ 648.014.

Average time, t = (t1 + t2)/2

Which gives;

t = (562.023 + 648.014)/2 ≈ 605.02

The average time it takes the man to escape in first move is t ≈ 605.02 seconds.

Learn more about Pythagorean theorem and probability here:

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