When Lisa first moved to Rose Valley one decade ago, it had a population of 82,400 people. Today, it has a population of 79,928 . Lisa heard on the news that the population of Rose Valley is expected to continue decreasing each decade. Write an exponential equation in the form \( y=a(b)^{x} \) that can model the population of Rose Valley, \( y \), x decades after Lisa moved there. Use whole numbers, decimals, or simplified fractions for the values of a and \( b \). \[ y=\square \text { (iD) } \] To the nearest hundred people, what can Lisa expect the population of Rose Valley to be 3 decades after moving there?

\[ \textbf{Step 1. Find the base } b. \] Lisa moved in when the population was \(82400\) (when \(x=0\)). One decade later (\(x=1\)), the population was \(79928\). In an exponential model of the form \[ y = a(b)^x, \] we have \(a = 82400\) and at \(x=1\) \[ 82400(b)^1 = 79928. \] Thus, \[ b = \frac{79928}{82400}. \] \[ \textbf{Step 2. Write the Exponential Equation.} \] The model is \[ y = 82400 \left(\frac{79928}{82400}\right)^x. \] \[ \textbf{Step 3. Find the Population 3 Decades Later (\(x=3\)).} \] Substitute \(x=3\) into the equation: \[ y(3) = 82400 \left(\frac{79928}{82400}\right)^3. \] \[ \textbf{Step 4. Approximate the Population.} \] Calculate the base: \[ \frac{79928}{82400} \approx 0.96963. \] Calculate its cube: \[ (0.96963)^3 \approx 0.91076. \] Then, \[ y(3) \approx 82400 \times 0.91076 \approx 75048. \] Rounded to the nearest hundred people, the population is approximately: \[ 75000. \] \[ \textbf{Final Answers:} \] Exponential model: \[ y = 82400 \left(\frac{79928}{82400}\right)^x. \] Population after 3 decades (rounded): \[ 75000. \]