College Algebra

Question

I need help understanding and breaking these questions down please...

1. In 1996, an outbreak of a disease infected 11 people in a large community. By 1997, the number of those infected had grown to 35.
2. A certain radioactive isotope decays at a rate of 0.1% annually. Determine the half-life of this isotope, to the nearest year.

Answer 1

So Shannon S.,

When you say, breaking down the problems, I'm assuming you mean "transforming them into appropriate mathematical models".

So, the first issue in problem 1 is, what kind of a process are we dealing with? Is the disease something which is being caught from rats (Yersina pestis - plague!), or from cats (Toxoplasmodium gondii -- a truly nasty organism, read up on it!), or exclusively from other people? And, if it is a communicable disease, is the rate of spread solely a linear function of the number of people who currently have it? (or have politicians started to get to work on it??)

If the latter is the case (especially if politicians are denying that there is a problem!), then you are well set up for exponential growth

N(t) = N₀e^kt where N(t) is the number of infected patients at time t (in years, starting at 1996 as t=0), N₀ is the infected count in 1996 (t=0), and k is the rate constant. By substituting in the known values, we can find:

N(t=1)= 35 = 11e^1k which solves for k= 1.15745 .

So now write the final equation: N(t) = 11e^{1.15745 t} .

Now obviously, you could also express that in terms of calendar years for t, but why bother.

Question 2 is somewhat similar, but here, you are definitely dealing with exponential decay, not growth. So the rate constant will act as a negative number, or you can write the equation with a negative sign in the exponent (that's what's usually done).

So the "at one year" condition is: (as an equation)

0.999 = 1e^-kt which solves as k= 0.0010005 .

Then you have to set up as:

0.5 = 1e^{-0.0010005 t}

That's t = 692.8 years, or 693 to the nearest year.

Note that that's a bit different than the "linear" guess from 0.1%/year up to 50% = 50%/0.1% yr^-1 = 500 years.

-- Cheers, --Mr. d.