GT Math Puzzler #2, December 10-14, 2012
1) The wind chill is the "perceived" temperature to the human body, based on both air
temperature and wind speed. On a cold, windy day, your body loses more heat than it loses
on a cold day with no wind. Heat is literally blown away from your body, causing you to feel
colder.
For examples, with a temperature of -15 degrees F and winds blowing at 35 mph, the wind
chill index would be -74 degrees F and would cause exposed skin to freeze in 30 seconds.
With the temperature of 15 degrees F and winds blowing at 35 mph, the wind chill index
would be -27 degrees F, which would likely cause frostbite and make outdoor activities
dangerous. Using just these numbers, what would you guess the wind chill index would be on
a day when the temperature is 45 degrees and winds blowing at 35 mph?
2) The formula for calculating the wind chill index is actually rather complex. When the wind
speed is v miles per hour and the temperature is t degrees Fahrenheit, the wind chill index
is given by the following formula:
wind chill index = 0.0817 (3.71v0.5 + 5.81 - 0.25v)(t - 91.4) + 91.4
Now, calculate the true wind chill index for a temperature of 45 degrees F and a wind speed
of 35 mph. How does this compare with the estimate you gave in the first question?
3)
The holidays are here, and many offices and classrooms to do a gift exchange. These often
work as follows: the names of all participants are put into a hat, and everyone draws a name.
Of course, if you draw your own name, you put it back and draw another. Let's say you work
in a twelve-person office, and all twelve people's names were put into a hat. Further, let's
say that you're the twelfth person to draw a name from the hat. It would be very bad if you
drew your own name, since returning it and drawing another would be impossible. What's the
probability that when you draw a name from the hat, you'll draw your own name?
4) In a four-person classroom, they decide to do a gift exchange as described above. What's
the likelihood that no one will choose their own name from the hat?
5) Can you predict the probability that no one will select his or her own name from the hat,
regardless of the number of people participating? That is, can you find a formula in terms
of n when n people participate in a gift exchange?