If interest in a bank account is compounded continuously, the amount grows at a rate that is proportional to the amount present in the account. Interest that is compounded daily very closely approximates this situation. Determine the amount in an account after one year if \( \$ 7000 \) is placed in account and it pays \( 4 \% \) interest per year, compounded continuously. After 1 year the account balance will be approximately \( \$ \square \) (Round to the nearest cent as needed.)

To determine the amount in the account after one year with continuous compounding, we can use the formula: \[ A = Pe^{rt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the time the money is invested for (in years). - \( e \) is the base of the natural logarithm (approximately equal to 2.71828). Plugging in the values: - \( P = 7000 \) - \( r = 0.04 \) - \( t = 1 \) \[ A = 7000 \times e^{0.04 \times 1} \approx 7000 \times e^{0.04} \] Calculating \( e^{0.04} \): \[ e^{0.04} \approx 1.04081 \] Thus, \[ A \approx 7000 \times 1.04081 \approx 7285.67 \] So after 1 year, the account balance will be approximately \( \$ 7285.67 \). --- This compounding concept has been a game-changer in finance! Continuous compounding means your money can grow at every conceivable moment, which is pretty thrilling if you're into maximizing savings. It’s fascinating to think that even the smallest amount can blossom into something more substantial if given enough time and compounding frequency. If you're new to the world of finance, diving into the math behind different compounding methods can be quite enlightening! You might want to explore how changing interest rates or compounding frequencies impacts your savings. A great book to consider is "The Mathematics of Money: Math and Financial Literacy" by James T. Wooten, which opens up the intricacies of personal finance while making the numbers approachable and fun!