Sure, Jemel wants to model the population growth of birds using the exponential formula [tex]\( a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial population, [tex]\( b \)[/tex] is the growth factor, and [tex]\( x \)[/tex] is the number of years.
Let's determine the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
1. Initial Population ( [tex]\(a\)[/tex] ):
- The initial population of birds is given as 240.
- Therefore, [tex]\( a \)[/tex] should be [tex]\( 240 \)[/tex].
2. Growth Factor ( [tex]\(b\)[/tex] ):
- The growth rate is given as 16% per year. To express this as a decimal, we divide by 100, so 16% becomes 0.16.
- The growth factor [tex]\( b \)[/tex] is calculated by adding 1 to the decimal growth rate. This accounts for both the existing population and its growth.
- Therefore, [tex]\( b = 1 + 0.16 = 1.16 \)[/tex].
So, Jemel should use the following values:
[tex]\[
a = 240
\][/tex]
[tex]\[
b = 1.16
\][/tex]
These values will correctly model the population growth of the birds after [tex]\( x \)[/tex] years using the given exponential formula.
