phillips exeter academy mathematics 1 answer key july 2014

Question

How long would it take you to count to one billion, reciting the numbers one after
another? First write a guess into your notebook, then come up with a thoughtful answer.
One approach is to actually do it and have someone time you, but there are more manageable
alternatives. What assumptions did you make in your calculations?

Answer 1

If the average amount of time to recite each number was approximately 2 seconds

Then it would take 2 billion seconds..

Noting that it takes a couple of seconds to say

" nine hundred ninety nine million, nine hundred ninety nine thousand, nine hundred and ninety nine"

 

2,000,000,000 seconds divided by 60 sec/minute, divided by 60 mins/hour, divided by 24 hours/day

comes out to 23,148 days.

 

This is approximately 63.4 years

 

This is without taking into account eating, sleeping or any other factors

which will extend this time period.

 

You can, instead of stating you count for 24 hours per day... say you count

for about 15 hours per day to allow for eating, sleeping

 

2,000,000,000/(60*60*15) = 37037 days

 

This is approximately 101.5 years

 

 

Answer 2

The process of estimating is important!  Many people have little idea of how to make and refine estimates. 

 

To "actually do it and have someone time you" initially sounds like a good method -- until you have spent days and are still counting relatively small numbers.  Now, you realize that planned vs. actual is important.  For a process that is repeated often (for example, mowing a particular lawn), recording times helps to create practical "expected values."

 

So, what about counting 1000 numbers somewhere in the middle range, say 36,542,334 - 36,543,334?  This time is multiplied by 1,000,000 to get an estimate for the problem.  This is called a "prototype" and is good to do with projects that you are attempting for the first time (for example, a new airplane is not flown until all the parts are systematically "broken" and the failure points recorded; this happens before the plane is built).

 

"How long" must be clarified.  The elapsed time is not as important as the active time (which does not count rest, meals, sleep, etc.).

 

Once you get an estimate, you may not want to do (or may not be able to do) the experiment any more.  The fundamental approach to security is to make it more difficult, more costly, more time-consuming ... than it is worth.