If a principal of P dollars is deposited in an account...

If a principal of $P$ dollars is deposited in an account earning interest at the rate of $r /$ year compounded continuously, then the accumulated amount at the end of $t$ years is given by $$A=f(P, r, t)=P e^{r t}$$ dollars. Find the accumulated amount at the end of 3 years if a sum of $\ 10,000$ is deposited in an account earning interest at the rate of $6 \% /$ year.

If a principal of $P$ dollars is deposited in an account earning interest at the rate of $r /$ year compounded continuously, then the accumulated amount at the end of $t$ years is given by $$A=f(P, r, t)=P e^{r t}$$
dollars. Find the accumulated amount at the end of 3 years if a sum of $\ 10,000$ is deposited in an account earning interest at the rate of $6 \% /$ year.

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Key Concepts

Continuous Compounding

This financial concept involves calculating interest where the interest accrues constantly, rather than at discrete intervals. The formula A = P * e^(rt) represents this process, where the accumulation depends on the initial principal, the rate, and the time period, and the exponential nature indicates continuous growth.

Exponential Function

An exponential function has the form f(x) = e^(kx) and is used to model processes where growth or decay happens at a continuous rate. In the context of continuous compounding, it mathematically captures the idea that interest is being added at every possible moment, leading to exponential growth over time.

Principal and Interest Rate

The principal is the initial amount deposited or invested, and the interest rate is the percentage at which the principal grows over time. In continuous compounding, these quantities interact with the time variable within an exponential function to determine the final accumulated amount.

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