Zombie Apocalypse Precal

Question

As we all know the zombie apocalypse starts with a single infected human. We found that 2 hours after the first infected there are 25 total infected. Assuming the population is not smart enough to effectively stop the spread of the zombie virus, and it continues to grow and an exponential rate (Bounded by the population of course) then how long before there are 1,000 infected? 50,000? 1 million? Everyone (7.9 billion)?

Round to two places.

1000 people: 

hours.50,000 people: 

 hours.1,000,000 people: 

 hours.7,900,000,000 people: 

 hours.

Answer 1

Autumn,

The basic form for exponential growth or decay is f(x)=a·b^x. It's exponential because x is in the exponent.

We have two data points. When we begin at time=0, the population infected is 1, which is point (0,1). When time=2, the population infected is 25. This is point (2,25).

You can use a regression function on a calculator or on Desmos.com to get the equation here, but maybe you see that the population increases by a factor of 5² between 0 and 2 hours. Therefore, using P for population and t for time, P(t)=1·5^t, or just P(t)=5^t.

Plug the populations in for P(t) and solve for t using logarithms to find the times it takes to get to the other populations:

1,000=5^t can be written in logarithmic form as log₅ 1000 = t. Use a calculator to get t=4.29.

Set P equal to the other values to get the time associated with them.

Finally, for 7,900,000,000=5^t, log₅ 7,900,000,000 = t. t = 14.16.

These people have not watched enough zombie movies!! They need better defenses! It doesn't even take 15 hours for zombies to take over the entire planet!! (Do they swim? Fly planes? How does this happen!?)

Woah.

I hope this helps. Let me know if you have questions, and have fun mathing!

Liz Z.