Question

Use the formula p=le^kt. A bacterial culture has an initial population of 500. If its population grows to 7000 in 2 hours, what will it be at the end of 4 hours

Answer 1

98,000 bacteria.

Explanation:

We are given the formula:

`P=le^(kt)`

Where l is the initial population, k is the rate of growth, and t is the time ( in hours).

We know that it has an initial population of 500. So, l is 500.

We also know that the population grows to 7000 after 2 hours.

And we want to find the population after 4 hours.

First, since we know that the population grows to 7000 after 2 hours, let's substitute 2 for t and 7000 for P. Let's also substitute 500 for l. This yields:

`7000=500e^(2k)`

Divide both sides by 500:

`e^(2k)=14`

We can solve for the rate k here, but this is not necessary. In fact, when can find our solution with just this.

Let's go back to our original equation. We want to find the population after 4 hours. So, substitute 4 for t:

`P=500e^(4k)`

We want to find the total population, P. Notice that we can rewrite our exponent as:

`P=500(e^(2k))^2`

This is the exact same thing we acquired earlier. So, we know that the expression within the parentheses is 14. Substitute:

`P=500(14)^2`

Square and multiply:

`P=500(196)=98000`

So, after 4 hours, there will be 98,000 bacteria.

And we're done!