The population of Salt Lake City, Utah, in 2010 was approximately 1,091,000. From 2010 to 2023, the population of Salt Lake City increased by 11.4%. If an exponential function f ca... The population of Salt Lake City, Utah, in 2010 was approximately 1,091,000. From 2010 to 2023, the population of Salt Lake City increased by 11.4%. If an exponential function f can be used to model the population, f(t), of Salt Lake City t years after 2010, which of the following equations best defines function f? Read more

Steps to Solve

1. Understanding Population Growth Function

   The general form of an exponential growth function is given by: $$ f(t) = P_0 \cdot (1 + r)^t $$ where ( P_0 ) is the initial population, ( r ) is the growth rate, and ( t ) is the time in years.

2. Setting Initial Population and Growth Rate

   From the problem, we know:

   • Initial population ( P_0 = 1,091,000 )
   • Growth rate ( r = 0.114 ) (which corresponds to a 11.4% increase).

3. Formulating the Function

   Based on the above, we can write the function for the population after ( t ) years: $$ f(t) = 1,091,000 \cdot (1 + 0.114)^t = 1,091,000 \cdot (1.114)^t $$

4. Analyzing the Options

   Now we compare this equation to the options provided:

   • Option A: ( f(t) = 1,091,000(1.114)^{t/13} ) → wrong, because of division by 13.
   • Option B: ( f(t) = 1,091,000(1.114)^t ) → this matches our derived function.
   • Option C: ( f(t) = 1,091,000(11.4)^{t/13} ) → wrong, bases must be ( 1.114 ).
   • Option D: ( f(t) = 1,091,000(1.114/13)^t ) → wrong, incorrect base.

5. Choosing the Correct Answer

   The correct answer is option B, which accurately models the population growth with a straight exponent.