Exponential Equations 17

Question

A population of bacteria is growing according to the equation  where t is measured in days. Estimate the number of days it will take for the population to exceed 2485.t =   

Answer 1

We need to find t when P(t) = 2485. Let's start by setting up the equation knowing this information:

2485 = 1300*e^0.09t (Equation 1)

Now, we want to isolate the exponential function on the right side of equation 1. Divide both sides by 1300.

2485 / 1300 = e^0.09t

Once that's done, we want to pull the t out of the exponential function. We can do that by taking the natural log of both sides:

ln(2485 / 1300) = ln( e^0.09t)

Since ln(e^n*t) becomes n*t by algebra, where n is a constant and t is our variable of the function, we can manipulate the equation like so:

ln(2485 / 1300) = 0.09*t

Finally, isolate the t by dividing both sides by 0.09. When isolated, you will get an answer with a decimal. To estimate, round to the nearest whole day. You want a number that will make P(t) exceed a value greater than 2485.

If 7.1989 were rounded to 7 days, the population would only be approximately 2440. This means that to exceed the population of 2485, it is best to round the 7 days to 1 day more, to get 8 days. This will give a population of approximately 2670, and 2670 exceeds 2485.

Therefore, t = 8 days

Therefore, the final answer becomes t = 7 days