Problem
In 2000 the population of country A reached 5 million, and in 2025 it is projected to be 6.5 million. (a) Find values for $P_{0}$ and a so that the following formula models the population of country $A$ in year $x$. \[ f(x)=P_{0} a^{x-2000} \] (b) Estimate the country's population in 2010 to the nearest hundredth of a million. (c) Use $\mathrm{f}$ to determine the year during which the country's population might reach 12 million. (a) Find values for $P_{0}$ and $a$. \[ \mathrm{P}_{0}=5 \text { million } \] (Round to the nearest hundredth as needed.) \[ a=1.01055 \] (Round to five decimal places as needed.) (b) The population in 2010 will be $\square$ million. (Use the rounded answer from part (a) to find this answer. Round to the nearest hundredth as needed.)
Solution
Step 1 ：Given that in 2000, the population was 5 million, we can say that \(P_{0} = 5\) million.
Step 2 ：In 2025, the population is projected to be 6.5 million. We can use these two points to solve for \(a\) in the formula \(f(x)=P_{0} a^{x-2000}\).
Step 3 ：By substituting the given values into the formula, we get \(6.5 = 5a^{2025-2000}\). Solving this equation gives us \(a = 1.01054983172935\).
Step 4 ：Rounding \(a\) to five decimal places, we get \(a = 1.01055\).
Step 5 ：Final Answer: The values for \(P_{0}\) and \(a\) are \(P_{0} = \boxed{5}\) million and \(a = \boxed{1.01055}\) (rounded to five decimal places).
