What was the population of the city in 2012?

a) Exponential Growth Function:

The general form of an exponential growth function is:

P(t) = P₀ x (1 + r)^t

where:

P(t) is the population at time t

P₀ is the initial population (5.74 million in this case)

r is the growth rate per year (3.75% = 0.0375)

t is the time in years

Therefore, the specific exponential growth function for this city is:

P(t) = 5.74 x (1 + 0.0375)^t

This function represents the population at any given year t after 2012.

b) Estimate the Population in 2018:

To estimate the population in 2018, we need to find P(6) since 2018 is 6 years after 2012. Plugging in the values:

P(6) = 5.74 x (1 + 0.0375)^6 ≈ 7.04 million

Therefore, the estimated population of the city in 2018 is approximately 7.04 million.

c) When will the Population Reach 8 Million?

We need to find the value of t for which P(t) = 8. Setting up the equation:

8 = 5.74 x (1 + 0.0375)^t

Solving for t using iterative methods or calculators, we get:

t ≈ 8.39 years

Since we only care about whole years, rounding up to 9 gives us the answer.

Therefore, the population of the city will reach 8 million in the year 2021 (2012 + 9 years).

d) Doubling Time:

The doubling time is the time it takes for the population to double its initial value. We can find it using the formula:

doubling time = ln(2) / ln(1 + r)

where ln is the natural logarithm. Plugging in the values:

doubling time ≈ 18.83 years

Therefore, it takes approximately 18.83 years for the city's population to double at its current growth rate