Question

How do you write an exponential equation for this problem

Answer 1

Let's think about the information in the problem. The problem tells us a few key points:

• The number of rabbits grows exponentially
• We start with 20 rabbits ($t = 0$, $a = 20$)
• After 6 months ($t = 6$), we have 100 rabbits ($a = 100$)

Since we know we are going to be working with an exponential model, we can start with a base exponential model:

$a = P \cdot r^t$

• $P$ is the principal, or starting amount
• $r$ is the growth/decay rate (in this case, growth)
• $t$ is the number of months
• $a$ is the number of rabbits

Based on the information in the problem, we can create two equations:

$20 = P \cdot r^0 = P$

$100 = P \cdot r^6$

The first equation tells us that $P = 20$, or that we start with 20 rabbits.  Thus, we can change the second equation to:

$100 = 20 \cdot r^6$

$5 = r^6$

Now, we don't know $r$, but we want to, so let's solve for it.

$5 = r^6$

$r = \sqrt[6]{5}$

Now, the problem is asking us how many rabbits we are going to have after one year ($t = 12$), so let's find that:

$a = 20 \cdot (\sqrt[6]{5})^{12}$

$a = 20 \cdot (5^{\frac{1}{6}})^{12}$

$a = 20 \cdot 5^2$

$a = 500$

After one year, we will have 500 rabbits.