In 2012 , the population of a city was 6.36 million. The exponential growth rate was $1.83 \%$ per year. a) Find the exponential growth function. b) Estimate the population of the city in 2018. c) When will the population of the city be 8 million? d) Find the doubling time.

Step 1 ：First, we need to understand the exponential growth function. The general form of an exponential growth function is \(P(t) = P_0(1 + r)^t\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time.

Step 2 ：For part a), we are asked to find the exponential growth function. We know that the initial population \(P_0\) is 6.36 million, and the growth rate \(r\) is 1.83%, or 0.0183 in decimal form. So, the exponential growth function is \(P(t) = 6.36(1 + 0.0183)^t\).

Step 3 ：For part b), we are asked to estimate the population of the city in 2018. This is 6 years after 2012, so \(t = 6\). Substituting \(t = 6\) into the exponential growth function, we get \(P(6) = 6.36(1 + 0.0183)^6\). Calculating this gives us approximately 7.36 million.

Step 4 ：For part c), we are asked to find when the population of the city will be 8 million. This means we need to solve the equation \(8 = 6.36(1 + 0.0183)^t\) for \(t\). Taking the natural logarithm of both sides and solving for \(t\) gives us approximately \(t = 13.5\) years. Since we can't have a fraction of a year, we round up to the nearest whole number, so the population will reach 8 million in 14 years, or in the year 2026.

Step 5 ：For part d), we are asked to find the doubling time. This means we need to solve the equation \(2 = (1 + 0.0183)^t\) for \(t\). Taking the natural logarithm of both sides and solving for \(t\) gives us approximately \(t = 38\) years. So, the population will double in 38 years.