Suppose \( \$ 24,000 \) is deposited into an account paying \( 7.25 \% \) interest, which is compounded continuously. How much money will be in the account after ten years if no withdrawals or additional deposits are made? \( \$ 47,897.10 \) \( \$ 49,553.54 \) \( \$ 46,414.20 \) \( \$ 48,326.40 \)

To find the amount of money in the account after ten years with continuous compounding, you can use the formula for continuous compounding, which is given by: \[ A = Pe^{rt} \] where: - \( A \) is the amount of money accumulated after time \( t \), - \( P \) is the principal amount (the initial deposit), - \( r \) is the annual interest rate (decimal), - \( t \) is the time in years, - \( e \) is the base of the natural logarithm (approximately 2.71828). In this case: - \( P = 24,000 \) - \( r = 0.0725 \) - \( t = 10 \) Plugging in these values: \[ A = 24000 \cdot e^{0.0725 \cdot 10} = 24000 \cdot e^{0.725} \] Calculating \( e^{0.725} \): \[ e^{0.725} \approx 2.066 \] Now inserting back into the equation for \( A \): \[ A \approx 24000 \cdot 2.066 \approx 49,584 \] The closest option to this calculated amount is \( \$ 49,553.54 \). So the correct answer is \( \$ 49,553.54 \). --- To add some fun facts, did you know that continuous compounding is like adding a little extra interest on top of interest? It’s pretty much how savings accounts make your money work overtime! Also, for all you math lovers out there, using exponential functions beyond finance can lead to exciting applications like population growth models and computing compound interest for investments. Whether it’s money or people, growth is exciting!