# 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 𝑛(𝑡) to describe the number of rabbits after 𝑡 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 𝑛 of 𝑡 to describe the number of rabbits after 𝑡 years.

In this question, we have our variables defined. We’re told that 𝑡 is the number of years, and 𝑛 of 𝑡 is the function that describes the number of rabbits. We know that since the population grows exponentially, we can use the equation d𝑃 by d𝑡 equals 𝑘 times 𝑃 to describe it. In fact though, we’ll rewrite this using our variables such that d𝑛 by d𝑡 equals 𝑘 times 𝑛. We could have written this using function notation such that 𝑛 prime of 𝑡 equals 𝑘 times 𝑛. But personally, I prefer to work with Leibniz notation.

Now, in fact, we have a value for 𝑘. We know that the relative growth rate is 23 percent. That’s 0.23 as a decimal. And we can therefore say that d𝑛 by d𝑡 equals 0.23 times 𝑛. Now remember, once d𝑛 by d𝑡 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 𝑛 d𝑛 equals 0.23 d𝑡. We’re then going to integrate both sides of this equation. The integral of the left-hand side is the natural log of 𝑛 plus some constant of integration 𝑎. And the integral of the right-hand side is 0.23𝑡 plus a different constant of integration 𝑏.

Now, note that usually the integral of one over 𝑛 would be the natural log of the absolute value of 𝑛. But we know the number of rabbits can’t be negative, so we don’t actually need to include that here. Combining our constants of integration by subtracting 𝑎 from both sides, and we see the natural log of 𝑛 is equal to 0.23𝑡 plus 𝑐. We then raise both sides to the power of 𝑒 to find that 𝑒 to the power of the natural log of 𝑛 is equal to 𝑒 to the power of 0.23𝑡 plus 𝑐. Well, 𝑒 to the natural log of 𝑛 is simply 𝑛.

And then, we use the laws of exponents to rewrite the right-hand side as 𝑒 to the power of 0.23𝑡 times 𝑒 to the power of 𝑐. But of course, 𝑒 to the power of 𝑐 is itself a constant. Let’s call it 𝑃. So, 𝑛 is equal to 𝑃 times 𝑒 to the power of 0.23𝑡. Now, we’re almost done. There were initially 245 rabbits on the farm. In other words, when 𝑡 is equal to zero, 𝑛 is equal to 245. So, 245 equals 𝑃 times 𝑒 to the power of zero because 𝑒 to the power of zero is one. So, we find 𝑃 to be equal to 245. And we found our function for 𝑛. We can say that 𝑛 of 𝑡 is equal to 245𝑒 the power of 0.23𝑡.