Answer:
[tex]y=3750 \cdot 1.0215^t[/tex]
(where [tex]y[/tex] is the population and [tex]t[/tex] is time in years after 2021)
5159
Step-by-step explanation:
Part (a)
General form of an exponential function: [tex]y=ab^x[/tex]
where:
If [tex]b > 1[/tex] then it is an increasing function
If [tex]0 < b < 1[/tex] then it is a decreasing function
We are told that the initial population is 3750. Therefore, [tex]a=3750[/tex]
We are told that the farm grows at a rate of 2.15% annually. Therefore, if it grows then every year it is 100% + 2.15% = 102.15% of the previous year.
Convert the percentage into a decimal:
102.15% = 102.15/100 = 1.0215
Therefore, [tex]b=1.0215[/tex]
We are told that the independent variable is [tex]t[/tex] (in years).
Therefore, the equation is [tex]y=3750 \cdot 1.0215^t[/tex]
(where [tex]y[/tex] is the population and [tex]t[/tex] is time in years after 2021)
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Part (b)
The year 2036 is 15 years after 2021. Therefore, substitute [tex]t = 15[/tex] into the equation and solve for [tex]y[/tex]:
[tex]\implies y=3750 \cdot 1.0215^{15}[/tex]
[tex]\implies y=5159.49068...[/tex]
Therefore, an estimate of the population of the town in 2036 is 5159.
