Finding a final amount in a word problem on exponential growth or decay A city has a population of 310,000 people. Suppose that each year the population grows by \( 6.5 \% \). What will the population be after 15 years? Use the calculator provided and round your answer to the nearest whole number. people

To find the population after 15 years, we can use the formula for exponential growth: \[ A = P(1 + r)^n \] where: - \( A \) is the final amount (population after 15 years), - \( P \) is the initial amount (population at the start, 310,000 people), - \( r \) is the growth rate (6.5% or 0.065), - \( n \) is the number of years (15 years). Substitute the given values into the formula: \[ A = 310,000(1 + 0.065)^{15} \] Now, we can calculate the population after 15 years using the calculator provided. Calculate the value by following steps: - step0: Calculate: \(310000\left(1+0.065\right)^{15}\) - step1: Add the numbers: \(310000\times 1.065^{15}\) - step2: Convert the expressions: \(310000\left(\frac{213}{200}\right)^{15}\) - step3: Simplify: \(310000\times \frac{213^{15}}{200^{15}}\) - step4: Rewrite the expression: \(200\times 1550\times \frac{213^{15}}{200^{15}}\) - step5: Reduce the numbers: \(1550\times \frac{213^{15}}{200^{14}}\) - step6: Rewrite the expression: \(50\times 31\times \frac{213^{15}}{200^{14}}\) - step7: Rewrite the expression: \(50\times 31\times \frac{213^{15}}{50^{14}\times 4^{14}}\) - step8: Reduce the numbers: \(31\times \frac{213^{15}}{50^{13}\times 4^{14}}\) - step9: Multiply the fractions: \(\frac{31\times 213^{15}}{50^{13}\times 4^{14}}\) The population after 15 years will be approximately 797,271 people. Rounding to the nearest whole number, the population after 15 years will be 797,271 people.