Exponential decay: Population problem concerning...

J

Jaskaran

Junior Member
Joined
May 5, 2006
Messages
67
Hey all,

I'm stumped on how to go about solving this particular problem.

"Scientists have developed a super-virus...the drug decreases the size of the population at a rate that is proportional to the number of survivors remaining. Given this information, how strong do scientists have to make the virus so that precisely 3 survivors remain at the end of 24 hrs? The initial population is 1,000."

I know that this concerns the general solution to dy/dt = ky

dy/y = k*dt

Integral (dy/y) = Integral (k*dt)

ln y = k*t + c

y = C * e^(kt)

And solving for k, I assume, if one wants to know how large k has to be so that 3 people remain at the end of 24hrs.

With that, how would I then figure out how many people remain "uncontaminated" after 10, 15, and 20 hrs?

Many, many, thanks!

-Cuanzo
 
Jaskaran said:
Hey all,

I'm stumped on how to go about solving this particular problem.

"Scientists have developed a super-virus...the drug decreases the size of the population at a rate that is proportional to the number of survivors remaining. Given this information, how strong do scientists have to make the virus so that precisely 3 survivors remain at the end of 24 hrs? The initial population is 1,000."

I know that this concerns the general solution to dy/dt = -ky

dy/y = k*dt

Integral (dy/y) = Integral (k*dt)

ln y = k*t + c

y = C * e^(kt)

And solving for k, I assume, if one wants to know how large k has to be so that 3 people remain at the end of 24hrs.

So you know how to solve for 'k' - I assume you did it. "C" is your initial population.

With that, your function is completly defined - now it is a matter of simple evaluation at a given value of 't'.

With that, how would I then figure out how many people remain "uncontaminated" after 10, 15, and 20 hrs?

Many, many, thanks!

-Cuanzo
Click to expand...
 
Thanks bruh.

I set it up as 1000e^k24 = 3

And get k = (ln3/1000)/24

Or do I have it wrongly set up?

Knowing k, I can definitely calculate what the population is at 10,15, or 20 hours.

So the value of k is -.242048.

But, interpreting in context, is this the "strength" of the virus? I know it's a proportionality constant, however
 
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