the initial population was 50 thousand people and the population increased by 12% each year. Determine the function rule for this definition.

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12-03-2018
Mathematics

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the initial population was 50 thousand people and the population increased by 12% each year. Determine the function rule for this definition.

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LeonardoVivaldi LeonardoVivaldi · Answer 1 · 2018-03-21T09:42:04+01:00

Answer: [tex]P(x) = 50,000(1.12)^x[/tex]

Explanation: Let [tex]P(x)[/tex] be the population after x years. Since 50,000 is the initial population, P(0) = 50,000. Moreover, since the population increases 12% per year,

[tex]P(x) = P(x - 1) + 12\% (P(x - 1)) \\ = P(x - 1) + 0.12(P(x - 1)) \\ \boxed{P(x) = 1.12(P(x - 1))}[/tex]

Using the preceding equation:

[tex]P(1) = 1.12(P(0)) = 50,000(1.12) \\ P(2) = 1.12(P(1)) = 1.12(50,000(1.12)) = 50,000(1.12)^2 \\ P(3) = 1.12(P(2)) = 1.12(50,000(1.12)^2) = 50,000(1.12)^3 \\. \\. \\. \\P(x) = 1.12(P(x-1)) = 1.12(50,000(1.12)^{x-1}) = 50,000(1.12)^x \\\boxed{P(x) = 50,000(1.12)^x}[/tex]