Suppose that 12,000 is invested in a bond fund and the...

Suppose that $\$ 12,000$ is invested in a bond fund and the account grows to $\$ 14,309.26$ in 4 yr. (See Example 2) a. Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding. b. How long will it take the investment to reach $\$ 20,000$ if the rate of return continues? Round to 1 decimal place.

Suppose that $\$ 12,000$ is invested in a bond fund and the account grows to $\$ 14,309.26$ in 4 yr. (See Example 2)
a. Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding.
b. How long will it take the investment to reach $\$ 20,000$ if the rate of return continues? Round to 1 decimal place.

Key Concepts

Continuous Compounding

Continuous compounding is the process of earning interest on an investment where the interest is added to the principal continuously rather than at discrete intervals. The mathematical model is represented by the equation A = P e^(rt), where A is the ending amount, P is the original principal, r is the annual interest rate, t is time in years, and e is Euler's number. This concept is fundamental in finance for modeling investments that accrue interest in a continuous manner, leading to exponential growth.

Exponential Growth

Exponential growth refers to the phenomenon where a quantity increases at a rate proportional to its current value, resulting in growth that accelerates over time. In finance, this is seen through the accumulation of interest on both the original principal and the accumulated interest, leading to an exponential increase in investment value. The continuous compounding formula is a practical application of exponential growth, highlighting how small, continuous increases can lead to significant growth over time.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms, which are the inverses of exponential functions. They are used to solve for variables appearing in the exponent, such as in determining the time required for an investment to reach a certain value or calculating the rate of return. By applying logarithms to an exponential equation, one can extract the variable of interest, making logarithms a critical tool in analyzing models based on continuous compounding.

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