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]
