Question

A town has a population of 13000 and grows at 4.5% every year. To the nearest tenth of a year, how long will it be until the population will reach 16900?

Answer 1

Answer:

Correct answer: n = 5.96 ≈ 6 years

Step-by-step explanation:

Given:

Currently population P = 13,000

The percentage annual population growth is 4.5 % or in decimal notation

p = 1.045

Population after n years  Pₙ = 16,900

Work:

After first year it will be   P₁ = P · p

After second year it will be   P₂ = P₁ · p = P · p²

After third year it will be   P₃ = P₂ · p² = P · p³

.....................................................................................

After n-th year it will be   Pₙ = P · pⁿ

pⁿ = Pₙ / P  

n = log p (Pₙ / P) = ln (Pₙ / P) / ln p

n = ln (16,900/ 13,000) / ln 1.045 = ln 1.3 / ln 1.045 = 5.96 ≈ 6

If we accept  n = 5.9 we will get:

Pₙ = P · pⁿ = 13.000 · 1.045⁵°⁹ = 16,855

If we accept  n = 5.96 we will get:

Pₙ = P · pⁿ = 13.000 · 1.045⁵°⁹⁶ = 16,899.6

If we accept  n = 6 we will get:

Pₙ = P · pⁿ =  P₆ = P · p⁶ = 13.000 · 1.045⁶ = 16,929.38

What you will accept is yours choice.

God is with you!!!

Answer 2

Answer: it will take 6 years until the population will reach 16900

Step-by-step explanation:

The growth rate is exponential. We would apply the formula for exponential growth which is expressed as

A = P(1 + r)^ t

Where

A represents the population after t years.

t represents the number of years.

P represents the initial population.

r represents rate of growth.

From the information given,

A = 16900

P = 13000

r = 4.5% = 4.5/100 = 0.045

Therefore

16900 = 13000(1 + 0.045)^t

16900/13000 = (1.045)^t

1.3 = (1.045)^t

Taking log of both sides to base 10

Log 1.3 = log1.045^t = tlog1.045

0.114 = 0.0191t

t = 0.114/0.0191

t = 6.0 years to the nearest tenth