A person deposits money into an account, which pays 6 %...

A person deposits money into an account, which pays $6 \%$ interest compounded continuously, at a rate of $\$ 1000$ per year for 30 years. Calculate: (a) The balance in the account at the end of the 30 years. (b) The amount of money actually deposited into the account. (c) The interest earned during the 30 years.

A person deposits money into an account, which pays $6 \%$ interest compounded continuously, at a rate of $\$ 1000$ per year for 30 years. Calculate:
(a) The balance in the account at the end of the 30 years.
(b) The amount of money actually deposited into the account.
(c) The interest earned during the 30 years.

Key Concepts

Time Value of Money

The time value of money is a fundamental concept in finance that recognizes money has a different value over time due to its earning potential. Money deposited earlier in an investment will grow more due to the accrual of interest compared to the same amount deposited later, which is captured in formulas for continuously compounded interest and annuities.

Integration in Financial Calculations

Integration is used in financial mathematics to aggregate the effects of continuously made deposits, each growing at the compounded rate over the remaining investment period. By integrating a function that represents the growth of each small contribution, one can compute the total accumulated value for continuous cash flows.

Continuous Annuities

A continuous annuity involves making deposits or payments at a constant rate over time rather than in discrete intervals. In finance, the future value of a continuous annuity is determined by integrating the continuously compounded contributions, reflecting the accumulation of each infinitesimal deposit as it earns interest over the remaining period of the investment.

Continuous Compounding

Continuous compounding is the process where interest is calculated and added to the principal continuously, rather than at discrete intervals. This results in the exponential growth of the investment, modeled mathematically with the formula A = Pe^(rt), where A is the amount, P is the initial principal, r is the interest rate, and t is time.

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