Solving Equations

Question

There are lots of ants in a kitchen. Each day half are exterminated. After x days, there are 1/32 of the original number of ants left. What equation leads to solving for the number of days(x) that have passed?

Answer 1

If P(t) is the population of ants on day t, the rate at which this decreases is proportional to the current ant population. This is exponential decay, so you have an equation of the form:

P(x) = C * exp(-kt)

where C is the initial number of ants (at day t=0) and k is a constant. Since you know P(x) = (1/32)*C, you can substitute to get:

(1/32)*C = C * exp(-kx)

Now C is nonzero [there are "lots of ants" ;) ], so you can cancel it from both sides, giving:

1/32 = exp(-kx)

There's not enough information in the problem to determine the value of k, but this seems to answer the question as worded. You could solve this for x using logarithms: x = 5*ln(2)/k.