The fox population in a certain region has an annual growth rate of \( 10 \% \) per year. In the year 2012, there were 21,000 foxes counted in the area. What is the fox population predicted to be in the year 2019? Round to the nearest fox. foxes

To find the predicted fox population in the year 2019, 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, - \( t \) is the number of years. Given: - Initial population in 2012, \( P_0 = 21,000 \) foxes, - Annual growth rate, \( r = 10\% = 0.10 \), - Year 2019 is 7 years after 2012, so \( t = 7 \). Substitute the values into the formula: \[ P(7) = 21,000 \times (1 + 0.10)^7 \] Now, we can calculate the predicted fox population in the year 2019. Calculate the value by following steps: - step0: Calculate: \(21000\left(1+0.1\right)^{7}\) - step1: Add the numbers: \(21000\times 1.1^{7}\) - step2: Convert the expressions: \(21000\left(\frac{11}{10}\right)^{7}\) - step3: Simplify: \(21000\times \frac{11^{7}}{10^{7}}\) - step4: Rewrite the expression: \(1000\times 21\times \frac{11^{7}}{10^{7}}\) - step5: Rewrite the expression: \(10^{3}\times 21\times \frac{11^{7}}{10^{7}}\) - step6: Reduce the numbers: \(21\times \frac{11^{7}}{10^{4}}\) - step7: Multiply: \(\frac{21\times 11^{7}}{10^{4}}\) - step8: Simplify: \(\frac{21\times 11^{7}}{10000}\) The predicted fox population in the year 2019 is approximately 40,923 foxes. Rounding to the nearest fox, the predicted population is 40,923 foxes.