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A town has a population of 2000 and grows at 2% every year. To the nearest year, how long will it be until the population will reach 3000 ?

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VirαtKσhliidn

Answer:

[tex]\large\boxed{\sf 25\ years }[/tex]

Step-by-step explanation:

Here it is given that the initial population of a town is 2,000 . And it increases at rate of 2% per year . We need to find out in what time the population will become 3,000 . As we know that ,

[tex]\sf\qquad\longrightarrow A = P\bigg\lgroup 1+\dfrac{R}{100}\bigg\rgroup^n [/tex]

where ,

  • A is the final population .
  • P is the initial population .
  • R is the rate of growth .
  • n is the number of years .

[tex]\\\red{\bigstar}\underline{\underline{\boldsymbol{ On \ substituting\ the\ respective\ values\ , }}}[/tex]

[tex]\\\sf\qquad\longrightarrow 3000 = 2000\bigg\lgroup 1+\dfrac{2}{100}\bigg\rgroup^n \\\\[/tex]

[tex]\sf\qquad\longrightarrow \dfrac{3000}{2000}=\lgroup 1 + 0.02\rgroup ^n \\\\ [/tex]

Now from Binomial Theorem , we know that ,

[tex]\\\sf\qquad\longrightarrow \boxed{\red{\sf (1+x)^n = 1+nx }} \\[/tex]

  • if x << 1 . Hence here 0.02 <<1 .

[tex]\\\sf\qquad\longrightarrow 1.5 = 1+n(0.02)\\\\ [/tex]

[tex]\sf\qquad\longrightarrow 1.5-1 = 0.02n \\\\[/tex]

[tex]\sf\qquad\longrightarrow 0.02n =0.5\\ \\ [/tex]

[tex]\sf\qquad\longrightarrow n =\dfrac{0.5}{0.02} \\\\[/tex]

[tex]\sf\qquad\longrightarrow \pink{ time (n) = 25\ years } [/tex]

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