You hear on the news that over the next 5 years, the inflation rate will skyrocket to \( 12 \% \). If today a new Blu-ray movie costs \( \$ 19.99 \), assuming continuous compounding, how much will that same disk cost in 5 years? Round your answer to the nearest cent (hundredth).

To find the cost of the Blu-ray movie in 5 years with continuous compounding, we can use the formula for continuous compounding: \[ A = P \times e^{r \times t} \] Where: - \( A \) is the amount after time \( t \), - \( P \) is the principal amount (initial cost), - \( r \) is the annual interest rate (inflation rate), - \( t \) is the time in years. Given: - Initial cost of the Blu-ray movie, \( P = \$19.99 \), - Inflation rate, \( r = 12\% = 0.12 \), - Time, \( t = 5 \) years. Substitute the values into the formula: \[ A = 19.99 \times e^{0.12 \times 5} \] Now, we can calculate the cost of the Blu-ray movie in 5 years using continuous compounding. Calculate the value by following steps: - step0: Calculate: \(19.99e^{0.12\times 5}\) - step1: Multiply the numbers: \(19.99e^{0.6}\) - step2: Convert the expressions: \(19.99e^{\frac{3}{5}}\) - step3: Convert the expressions: \(\frac{1999}{100}e^{\frac{3}{5}}\) - step4: Multiply the numbers: \(\frac{1999e^{\frac{3}{5}}}{100}\) - step5: Simplify: \(\frac{1999\sqrt[5]{e^{3}}}{100}\) The cost of the Blu-ray movie in 5 years with continuous compounding is approximately $36.42. Rounding to the nearest cent, the cost will be $36.42.