Exponential Growth and Decay - Problem 3

So exponential growth occurs when things are increasing in number. So for example what we're going to be looking at is we're dealing with rabbits, these rabbits have a reputation of being a little promiscuous so the number of rabbits is going to grow exponentially.

So for this problem what we have is the number of rabbits on a farm increases according to this formula where our time is in months and we're asked three different parts about this problem.

First is the initial amount and the initial amount occurs when our time is 0. So we plug in t is equal to 0, and we end up with either the 0 anything to the 0 is 1 so this is just going to be n of t is equal to 200, we start with 200 rabbits.

How many rabbit after 10 months? Very, very similar in process to this one except in this case instead of having t is 0, we now have t is 10, simply plug in 10 and if 10 is equal to 200e to the point 3 times 10, plug this in our calculator 200e to the point 3 times 10 is just 3 giving us those are a lot of rabbits 4017 they really took off.

And our last one is how now to triple. So if we started with 200 rabbits, to have them triple that means we're going to have 3 times a many or 600. So we want 600 rabbits and that is going to equal to our equation 200e to the point 3t.

Now we just have to solve for t, solving the exponential equation always need to isolate our exponential divide by 200 leaving us with 3 is equal to e to the .3t, we need to get our variable down, so we need to take the log of both sides, natural log works best because it agrees with e, it give me so natural log of 3 is equal to .3t natural log of e. The natural log of e is just 1 to solve this out divided by point 3 leaving us with natural log of 3 over .3. Plug this into our calculator. The natural log of 3 divided by .3 is 3 point 66 and our units here is months.

So as you can see what happens with exponential growth is things start off a little bit slow but then they just sort of take off, so after 3 and a half months we only have 600 rabbits, check out what happens over the next 6 and a half, we go up to over 4000.

So exponential growth, the ingredients can change, but these steps figuring out what's going on with them is pretty much the same no matter what.