Using the Population Growth Formula

Added: 2026-05-28

[00:00:00]Now, the problem says we start out with 100. We grow at a rate proportional to our size. After an hour, the population gets to 420. How many bacteria are there after t hours? That's what we just figured out the answer to right here.

[00:00:12]Um, what is the rate of growth after 3 hours? So, let's go over here and we're going to write this equation down again.

[00:00:21]X of T is 100. Then we're going to have 21 over 5 to the^ of T. This is the population size. Now it's we want to know now what the rate of growth is. So x prime of t. How do we take the derivative of this thing? Well, this is a base. It's a weird fractional base, but it's just a number to the power of t. We know how to take derivatives of that. Uh first of all, the exponential remains indestructible. 21 over 5 to the t, right? But then we have to multiply by the natural log of the base. So 21 over 5. This is something that we've learned in the past. How to take derivatives of general exponential functions. The exponential function itself remains intact. And then we have the natural log of the base. There is another constant right there. All right.

[00:01:06]So this is the rate of change. So if we have x prime and we're trying to find out what it is at 3 hours in the future.

[00:01:12]Let me double check. What is the rate of growth after 3 hours? If we put three hours in here, we're going to get 100.

[00:01:19]Then we're going to get 21 over 5 to the^ of 3. Then we have natural log 21 over 5. So let me double check myself.

[00:01:27]100 21 over 5 to the^ 3 natural log of 21 over 5. And this works out to 10632.25.

[00:01:37]And that's bacteria per hour. Learn anything at math and science.com.