Use the compound interest formulas A = P (1+ r/n)^nt and A =Pe^rt to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is d. compounded continuously.

Verified step by step guidance

1. Identify the given values: principal P = 10,000, time t = 5 years, interest rate r = 1.32% (which is 0.0132 in decimal form).
2. For parts a, b, and c, use the compound interest formula $A=P(1+r/n){}^n$, where n is the number of compounding periods per year.
3. Calculate the accumulated value for each compounding frequency: semiannually (n=2), quarterly (n=4), and monthly (n=12) by substituting the values of P, r, n, and t into the formula.
4. For part d, use the continuous compounding formula $A=P{e}^r$, substituting the values of P, r, and t.
5. After substituting the values in each formula, compute the powers and products, then round each accumulated value to the nearest cent.

Key Concepts

Compound Interest Formula

The compound interest formula A = P(1 + r/n)^(nt) calculates the accumulated amount A after t years, where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is time in years. It accounts for interest earned on both the initial principal and the accumulated interest.

Continuous Compounding Formula

Continuous compounding uses the formula A = Pe^(rt), where e is Euler's number (~2.718), to calculate the accumulated amount when interest is compounded an infinite number of times per year. This formula models the limit of compound interest as compounding frequency increases indefinitely.

Rounding and Financial Precision

Rounding to the nearest cent means expressing the final amount to two decimal places, reflecting standard currency format. This ensures practical and accurate financial reporting, especially important when dealing with money and interest calculations.