Question

A town has a population of 14,000 and grows at 4.5% every year. To the nearest year, how long will it be until the population reaches 24,600?

a) 13 years

b) 15 years

c) 17 years

d) 19 years

Answer 1

The population of the town will reach 24,600 in approximately 13 years. Option A is correct.

Step-by-step explanation:

To find out how long it will take for the population of the town to reach 24,600, we can set up an equation using the exponential growth formula:

P = P0(1 + r)t

Where:
P is the final population (24,600)
P0 is the initial population (14,000)
r is the growth rate (4.5% or 0.045 as a decimal)
t is the time in years

Plugging in the values, the equation becomes:

24,600 = 14,000(1 + 0.045)t

To solve for t, we need to isolate it. Dividing both sides of the equation by 14,000 gives:

24,600 / 14,000 = (1 + 0.045)t

Simplifying the left side gives:

1.75 = (1 + 0.045)t

To solve for t, we take the logarithm (base 10) of both sides:

log(1.75) = log((1 + 0.045)t)

Using logarithm properties, we can bring the exponent down:

log(1.75) = t * log(1 + 0.045)

Now, we can solve for t by dividing log(1.75) by log(1 + 0.045) using a calculator:

t ≈ 13 years (rounded to the nearest year)