The balance owed on your credit card triples from \( \$ 500 \) to \( \$ 1500 \) in 9 months. If the balance is growing linearly then it would take 58.5 months to reach \( \$ 7000 \). If, on the other hand, the balance is growing exponentially, \( \mathrm{f}(\mathrm{x})=500(1+0.13)^{\mathrm{x}} \) where \( x \) represents the number of months, what would the balance be after 58.5 months? Round your answer to the nearest cent. AnswerHow to enter your answer (opens in new window) 2 Points

To find the balance after 58.5 months using the exponential growth model \( f(x) = 500(1 + 0.13)^x \), we substitute \( x = 58.5 \): \[ f(58.5) = 500(1 + 0.13)^{58.5} \approx 500(1.13)^{58.5} \] Calculating \( (1.13)^{58.5} \) gives approximately \( 59.3903 \). So, \[ f(58.5) \approx 500 \times 59.3903 \approx 29695.15 \] Thus, the balance after 58.5 months is approximately **$29695.15** when rounded to the nearest cent. Now, regarding this exponential growth pattern, it's fascinating to see how small increases can lead to dramatic effects over time, especially with financial instruments. In real life, understanding exponential growth helps in planning for investments, loans, and savings, emphasizing the importance of managing financial products wisely. Also, this concept closely links to real-world scenarios like population growth or viral sensations on social media, where things can seem moderate at first but then explode sharply. Fun fact: many folks underestimate how quickly things can grow exponentially, which is why budgeting and forecasting are key skills in finance and business!