Let's think about the information in the problem. The problem tells us a few key points:
Since we know we are going to be working with an exponential model, we can start with a base exponential model:
[tex]a = P \cdot r^t[/tex]
Based on the information in the problem, we can create two equations:
[tex]20 = P \cdot r^0 = P[/tex]
[tex]100 = P \cdot r^6[/tex]
The first equation tells us that [tex]P = 20[/tex], or that we start with 20 rabbits. Thus, we can change the second equation to:
[tex]100 = 20 \cdot r^6[/tex]
[tex]5 = r^6[/tex]
Now, we don't know [tex]r[/tex], but we want to, so let's solve for it.
[tex]5 = r^6[/tex]
[tex]r = \sqrt[6]{5}[/tex]
Now, the problem is asking us how many rabbits we are going to have after one year ([tex]t = 12[/tex]), so let's find that:
[tex]a = 20 \cdot (\sqrt[6]{5})^{12}[/tex]
[tex]a = 20 \cdot (5^{\frac{1}{6}})^{12}[/tex]
[tex]a = 20 \cdot 5^2[/tex]
[tex]a = 500[/tex]
After one year, we will have 500 rabbits.
An exponential function can represent growth or decay
You will have 511 rabbits at the end of one year
Represent the months with x, and the number of rabbits with y.
So, the parameters can be represented as:
[tex](x,y) = \{(0,20)\ (6,100)\}[/tex]
An exponential function is represented as:
[tex]y = ab^x[/tex]
At (0,20), we have:
[tex]ab^0 = 20[/tex]
Evaluate the exponent
[tex]a(1) = 20[/tex]
[tex]a = 20[/tex]
At (6,100), we have:
[tex]ab^6 = 100[/tex]
Substitute 20 for (a)
[tex]20b^6 = 100[/tex]
Divide both sides by 20
[tex]b^6 =5[/tex]
Take 6th roots of both sides
[tex]b =1.31[/tex]
So, the exponential function [tex]y = ab^x[/tex] becomes
[tex]y = 20(1.31)^x[/tex]
At the end of 1 year, the value of x is 12.
So, we have:
[tex]y = 20(1.31)^{12[/tex]
Evaluate
[tex]y = 511[/tex]
Hence, you will have 511 rabbits at the end of one year
Read more about exponential functions at:
https://brainly.com/question/11464095
