1. ## Exponential Growth and Decay: Deadly Virus!

So, there's this reality TV show that puts 1,000 people on an island. A super virus is unleashed on the island. This virus is designed to render victims lifeless for 24 hours (they're fine after that). The last 3 people who don't catch the bug wins a million dollars.

The scientists who developed the virus says that the drug decreases the size of the population at a rate that is proportional to the number of survivors remaining. With this information, how strong do the scientists have to make the virus so that only 3 survivors remain at the end of 24 hours? What does the "constant of proportionality" have to be? How many people can the audience expect to remain uncontaminated after 10, 15, and 20 hours?

So, I think I'm supposed to use P=ce^(kt)
t=24 hours
P=3
c=1000

But I'm extremely stumped on how to figure out the rest of this problem. I would appreciate any help. Thank you!

2. So I'll follow through with $\displaystyle P=ce^{kt}$, plugging in the numbers that you listed you can solve for k. Here's all the steps:

$\displaystyle 3=1000e^{24k}$

$\displaystyle 0.003 = e^{24k}$

$\displaystyle ln(0.003) = ln(e^{24k})$

$\displaystyle ln(0.003)=(24k)ln(e)$

$\displaystyle \frac{ln(0.003)}{24} = k$

$\displaystyle k = -0.242$

The formula then is $\displaystyle P=1000e^{-0.242t}$
Input in the times of 10, 15, and 20 hours to solve for P (number of survivors) in each case.

Hopefully that is all correct!