Expert: Paul Klarreich - 10/4/2005

Question
1.Suppose a disease is transmitted from individual to two others during the period of a week. Each of those two gives the disease to two more people in the second week, so that the total of newly infected people is now four. If the spread of the disease continues, how many weeks will it take a city of 300,000 people to have all been infected?

2.Use the growth and decay formula to solve the following problem:
Zebra mussels from Europe began invading the Mississippi River in 1988. In 1997 a portion of the river contained an average of 10 zebra mussels per square mile. The exponential growth rate was 340% (k=3.4). Predict the number of mussels per square mile in 2002


Answer
Hi, Nikki,
I answered this for Casey --perhaps the two of you could get together and send one question.

Hi, Casey,
Unfortunately, for the second question, I need more information -- exactly what is your 'growth and decay formula', including the definitions of the constants contained in it.

However, for the first one:

1.Suppose a disease is transmitted from a individual to two others during the period of a week. Each of those two gives the disease to two more people in the second week, so that the total of newly infected people is now four. If the spread of the disease continues, how many weeks will it take a city of 300,000 people to have all been infected?

In the first week, there is 1 person infected.
In the second, there are 1 + 2
In the third, there are 1 + 2 + 2^2.
In week 'n', there are 1 + 2 + 2^2 + ... 2^(n - 1)

That is a geometric series, with r = 2, whose sum is

S = 2^n - 1

Set 2^n - 1 = K (in this case 300000) and solve:

2^n = K - 1.
Now 2^n = exp(n log 2)

So exp(n log 2) = K - 1
Then n log 2 = log(K - 1)
log(k - 1)
and n = ----------
log 2

All logs are natural logs, of course.