Question

the population of a city doubled from 1950 to 1988, going from 2.5 million to 5 million people. using the exponential model, $p = p_0e^{rt}$, where $p$ is population, $p_0$ is initial population, and $t$ is time in years, find the annual growth rate $r$ for that period. although the annual growth rate has declined slightly to 1.52% annually, the population of the city is still growing at a tremendous rate. using the initial population of 5 million in 1988 and an annual rate of 1.52%, estimate the population of the city in the year 2010.

the annual growth, $r$, between 1950 and 1988 was 1.82 %.
(round to two decimal places as needed.)

the population of the city in the year 2010 should be about \\(\square\\) million people.
(round to one decimal place as needed.)

Explanation:

Step1: Determine time period

From 1988 to 2010, the number of years \( t = 2010 - 1988 = 22 \) years.

Step2: Identify initial population and growth rate

Initial population \( P_0 = 5 \) million, annual growth rate \( r = 1.52\% = 0.0152 \) (in decimal form).

Step3: Apply exponential growth formula

The exponential growth model is \( P = P_0 e^{rt} \). Substitute the values: \( P = 5 \times e^{0.0152 \times 22} \).
First, calculate the exponent: \( 0.0152 \times 22 = 0.3344 \).
Then, calculate \( e^{0.3344} \approx 1.396 \).
Finally, \( P = 5 \times 1.396 = 6.98 \approx 7.0 \) (rounded to one decimal place).

Answer:

7.0