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.