9. The population of Sacramento is currently 567,345 people. The population has been increasing by \( 0.31 \% \) annually. a) Write the exponential function b) What will the population be in 10 years? Round your answer to the nearest person.

To find the exponential function that represents the population of Sacramento, we can use the formula for exponential growth: \[ P(t) = P_0 \times (1 + r)^t \] where: - \( P(t) \) is the population at time \( t \), - \( P_0 \) is the initial population, - \( r \) is the annual growth rate as a decimal, - \( t \) is the time in years. Given: - Initial population \( P_0 = 567,345 \) people, - Annual growth rate \( r = 0.31\% = 0.0031 \) as a decimal. a) The exponential function representing the population of Sacramento is: \[ P(t) = 567,345 \times (1 + 0.0031)^t \] b) To find the population in 10 years, we substitute \( t = 10 \) into the function: \[ P(10) = 567,345 \times (1 + 0.0031)^{10} \] Let's calculate this. Calculate the value by following steps: - step0: Calculate: \(567345\left(1+0.0031\right)^{10}\) - step1: Add the numbers: \(567345\times 1.0031^{10}\) - step2: Convert the expressions: \(567345\left(\frac{10031}{10000}\right)^{10}\) - step3: Simplify: \(567345\times \frac{10031^{10}}{10000^{10}}\) - step4: Rewrite the expression: \(5\times 113469\times \frac{10031^{10}}{10000^{10}}\) - step5: Rewrite the expression: \(5\times 113469\times \frac{10031^{10}}{625^{10}\times 16^{10}}\) - step6: Rewrite the expression: \(5\times 113469\times \frac{10031^{10}}{5^{40}\times 16^{10}}\) - step7: Reduce the numbers: \(113469\times \frac{10031^{10}}{5^{39}\times 16^{10}}\) - step8: Multiply the fractions: \(\frac{113469\times 10031^{10}}{5^{39}\times 16^{10}}\) The population of Sacramento in 10 years will be approximately 585,180 people, rounded to the nearest person.