Question

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 ?

Answer 1

`\large\boxed{\sf 25\ years }`

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 ,

`\sf\qquad\longrightarrow A = P\bigg\lgroup 1+(R)/(100)\bigg\rgroup^n`

where ,

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

`\\\red{\bigstar}\underline{\underline{\boldsymbol{ On \ substituting\ the\ respective\ values\ , }}}`

`\\\sf\qquad\longrightarrow 3000 = 2000\bigg\lgroup 1+(2)/(100)\bigg\rgroup^n \\\\`

`\sf\qquad\longrightarrow (3000)/(2000)=\lgroup 1 + 0.02\rgroup ^n \\\\`

Now from Binomial Theorem , we know that ,

`\\\sf\qquad\longrightarrow \boxed{\red{\sf (1+x)^n = 1+nx }} \\`

• if x << 1 . Hence here 0.02 <<1 .

`\\\sf\qquad\longrightarrow 1.5 = 1+n(0.02)\\\\`

`\sf\qquad\longrightarrow 1.5-1 = 0.02n \\\\`

`\sf\qquad\longrightarrow 0.02n =0.5\\ \\`

`\sf\qquad\longrightarrow n =(0.5)/(0.02) \\\\`

`\sf\qquad\longrightarrow \pink{ time (n) = 25\ years }`