A town's population grows at the rate of 4% per year.If this growth rate remains constant, how long will it take the population to double?
A) $7.5$ years
B) $0.6$ years
C) $1.4$ years
D) $17.3$ years
E) $2.8$ years

Question

A town's population grows at the rate of 4% per year.If this growth rate remains constant, how long will it take the population to double? 
A) $7.5$ years
B) $0.6$ years
C) $1.4$ years
D) $17.3$ years
E) $2.8$ years

Quizplus · Accepted Answer

Let P be the initial population, then the population after t years is given by:

P(t) = P(0) * (1 + r)^t

where r is the annual growth rate as a decimal.

We want to find the time t it takes for the population to double, which means we want:

P(t) = 2 * P(0)

Substituting in the formula for P(t), we get:

P(0) * (1 + r)^t = 2 * P(0)

Dividing both sides by P(0), we get:

(1 + r)^t = 2

Taking the natural logarithm of both sides, we get:

ln(1 + r)^t = ln 2

Using the property of logarithms that ln(a^b) = b * ln(a), we can rewrite this as:

t * ln(1 + r) = ln 2

Solving for t, we get:

t = ln 2 / ln(1 + r)

Plugging in r = 0.04, we get:

t = ln 2 / ln(1.04) ≈ 17.3 years

Therefore, the answer is D) 17.3 years.

A Town's Population Grows at the Rate of 4% Per 7.57.57.5

A Town's Population Grows at the Rate of 4% Per $7.5$