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A population of 240 birds increases at a rate of [tex]\(16 \%\)[/tex] annually. Jemel writes an exponential function [tex]\(a \cdot b^x\)[/tex] to represent the number of birds after [tex]\(x\)[/tex] years. Which values should she use for

[tex]\[a = \][/tex]
[tex]\[b = \][/tex]

Sagot :

GinnyAnsweridn
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.
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