# Video: Applications of Exponential Growth

The population of rabbits on a farm grows exponentially. If there are currently 245 rabbits and the relative growth rate is 23%, find a function π(π‘) to describe the number of rabbits after π‘ years.

02:51

### Video Transcript

The population of rabbits on a farm grows exponentially. If there are currently 245 rabbits and the relative growth rate is 23 percent, find a function π of π‘ to describe the number of rabbits after π‘ years.

In this question, we have our variables defined. Weβre told that π‘ is the number of years, and π of π‘ is the function that describes the number of rabbits. We know that since the population grows exponentially, we can use the equation dπ by dπ‘ equals π times π to describe it. In fact though, weβll rewrite this using our variables such that dπ by dπ‘ equals π times π. We could have written this using function notation such that π prime of π‘ equals π times π. But personally, I prefer to work with Leibniz notation.

Now, in fact, we have a value for π. We know that the relative growth rate is 23 percent. Thatβs 0.23 as a decimal. And we can therefore say that dπ by dπ‘ equals 0.23 times π. Now remember, once dπ by dπ‘ is absolutely not a fraction, there are certain circumstances where we treat it a little like one. Here, we can rewrite our differential equation as one over π dπ equals 0.23 dπ‘. Weβre then going to integrate both sides of this equation. The integral of the left-hand side is the natural log of π plus some constant of integration π. And the integral of the right-hand side is 0.23π‘ plus a different constant of integration π.

Now, note that usually the integral of one over π would be the natural log of the absolute value of π. But we know the number of rabbits canβt be negative, so we donβt actually need to include that here. Combining our constants of integration by subtracting π from both sides, and we see the natural log of π is equal to 0.23π‘ plus π. We then raise both sides to the power of π to find that π to the power of the natural log of π is equal to π to the power of 0.23π‘ plus π. Well, π to the natural log of π is simply π.

And then, we use the laws of exponents to rewrite the right-hand side as π to the power of 0.23π‘ times π to the power of π. But of course, π to the power of π is itself a constant. Letβs call it π. So, π is equal to π times π to the power of 0.23π‘. Now, weβre almost done. There were initially 245 rabbits on the farm. In other words, when π‘ is equal to zero, π is equal to 245. So, 245 equals π times π to the power of zero because π to the power of zero is one. So, we find π to be equal to 245. And we found our function for π. We can say that π of π‘ is equal to 245π the power of 0.23π‘.