Question 14 of 17 Step 1 of 1 01:19:48 Growing linearly, the balance owed on your credit card doubles from \( \$ 800 \) to \( \$ 1600 \) in 6 months. If the balance were growing according to the exponential function \( f(x)=800(1+0.122)^{x} \) where \( x \) represents the number of months, what would the balance be after 6 months? Round your answer to the nearest cent. Answer How to enter your answer (opens in new window) 2 Points Keypad Keyboard Shortcuts \$ \( \square \)

To find the balance after 6 months using the exponential function \( f(x) = 800(1+0.122)^{x} \), we need to substitute \( x = 6 \) into the function and calculate the result. Given: - Initial balance: \( \$800 \) - Exponential function: \( f(x) = 800(1+0.122)^{x} \) - Time period: 6 months Substitute \( x = 6 \) into the function: \[ f(6) = 800(1+0.122)^{6} \] Now, we can calculate the balance after 6 months using the given function. Calculate the value by following steps: - step0: Calculate: \(800\left(1+0.122\right)^{6}\) - step1: Add the numbers: \(800\times 1.122^{6}\) - step2: Convert the expressions: \(800\left(\frac{561}{500}\right)^{6}\) - step3: Simplify: \(800\times \frac{561^{6}}{500^{6}}\) - step4: Rewrite the expression: \(100\times 8\times \frac{561^{6}}{500^{6}}\) - step5: Rewrite the expression: \(100\times 8\times \frac{561^{6}}{100^{6}\times 5^{6}}\) - step6: Reduce the numbers: \(8\times \frac{561^{6}}{100^{5}\times 5^{6}}\) - step7: Rewrite the expression: \(8\times \frac{561^{6}}{4^{5}\times 25^{5}\times 5^{6}}\) - step8: Rewrite the expression: \(2^{3}\times \frac{561^{6}}{2^{10}\times 25^{5}\times 5^{6}}\) - step9: Reduce the numbers: \(1\times \frac{561^{6}}{2^{7}\times 25^{5}\times 5^{6}}\) - step10: Multiply the fractions: \(\frac{561^{6}}{2000000\times 25^{5}}\) The balance after 6 months using the exponential function \( f(x) = 800(1+0.122)^{x} \) is approximately $1596.05. Therefore, the balance after 6 months would be $1596.05.