Route Roundabout

Geometry Level 3

You stumble on a roundabout.
You believe without a doubt
there is no more reason to pout.
It makes you want to scream and shout.

You and ride and ride and ride and ride.
To the thrill you now abide.
Your heart flung open wide and wide.

Round and round you feel sublime;
Went 50 to 100 times.

You forgot, but feel no shame:
Each possibility's the same.

Assume the radius is 1.
How far might you have had your fun?

Details and assumptions :

You travel on a fixed track around the circumference; don't add the distance it would take to turn upon exiting.
Round your answer to the tenth's place.
There is one exit on each of the four corners of the circle.

Inspired by riding my bicycle around the roundabout in the picture.

Satellite image was taken by Google. No copyright infringement intended.

The answer is 471.2.

1 solution

Brock Brown
Jul 22, 2015

Let $d$ be the expected distance that you travelled.

Let $\tau = 2\pi$ (see tau manifesto )

Method 1:

Because the chances are uniformly distributed, we can take the maximum possible distance and average it with the minimum possible distance .

$d=\frac{50\tau + 100\tau}{2} = 75\tau \approx \boxed{471.2}$

Method 2:

Python 3.3:

from random import randint
from math import pi
from time import time
from pickle import load, dump
def whee():
    distance = 2*pi*(randint(50*4, 100*4)/4)
    return distance
try:
    total, trials = load(open('roundabout.p', 'rb'))
except FileNotFoundError:
    total, trials = 0, 0
end = int(input("How long? (seconds) ")) + time()
while time() < end:
    total += whee()
    trials += 1
answer = round(total/trials, 1)
print ("d is around", answer)
# save results for later
dump((total, trials), open('roundabout.p', 'wb'))

$\implies d \approx \boxed{471.2}$

I prefer your second method!

Pi Han Goh - 5 years, 10 months ago

Me too. But the commoners who can't understand how to control snakes demand "mathematical rigor".

Brock Brown - 5 years, 10 months ago

Amateurs. "mathematical rigor" is just a bunch of random symbols put together.