Use the model A=P e^r t . The variable A represents the...

Use the model $A=P e^{r t} .$ The variable $A$ represents the future value of $P$ dollars invested at an interest rate $r$ compounded continuously for $t$ years. If a couple has $\$ 80,000$ in a retirement account, how long will it take the money to grow to $\$ 1,000,000$ if it grows by $6 \%$ compounded continuously? Round to the nearest year.

Use the model $A=P e^{r t} .$ The variable $A$ represents the future value of $P$ dollars invested at an interest rate $r$ compounded continuously for $t$ years.
If a couple has $\$ 80,000$ in a retirement account, how long will it take the money to grow to $\$ 1,000,000$ if it grows by $6 \%$ compounded continuously? Round to the nearest year.

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Key Concepts

Natural Logarithm

The natural logarithm is the inverse function of the exponential function with base e. It is used to solve equations where the variable is in the exponent, such as in the continuous compounding formula A = P e^(rt). By taking the natural logarithm of both sides of the equation, one can isolate the time variable and solve for it, making it a vital tool in analyzing exponential growth models.

Exponential Growth

Exponential growth describes a process in which the quantity increases at a rate proportional to its current value. In the context of finance, when interest is compounded continuously, the growth of an investment follows an exponential model. This concept is critical for understanding how investments increase exponentially over time under a constant interest rate.

Continuous Compounding

Continuous compounding refers to the process of calculating interest by assuming that interest is added to the principal an infinite number of times per period, effectively at every moment. This is modeled by the exponential function where the future value of an investment is given by A = P e^(rt), and it provides a more precise measure of growth compared to traditional periodic compounding.

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