An investment grows exponentially under continuous...

An investment grows exponentially under continuous compounding. After 2 yr, the amount in the account is \$7328.70. After 5 yr, the amount in the account is \$8774.10. Use the model $A(t)=P e^{r t}$ to a. Find the interest rate $r$. Round to the nearest percent. b. Find the original principal $P$. Round to the nearest dollar. c. Determine the amount of time required for the account to reach a value of $\$ 15,000 .$ Round to the nearest year.

An investment grows exponentially under continuous compounding. After 2 yr, the amount in the account is \$7328.70. After 5 yr, the amount in the account is \$8774.10. Use the model $A(t)=P e^{r t}$ to
a. Find the interest rate $r$. Round to the nearest percent.
b. Find the original principal $P$. Round to the nearest dollar.
c. Determine the amount of time required for the account to reach a value of $\$ 15,000 .$ Round to the nearest year.

Key Concepts

Logarithmic Transformation

Logarithmic transformation is a mathematical technique used to solve equations involving exponential functions. By applying the natural logarithm (ln) to an exponential equation, one can linearize the equation, making it possible to isolate and solve for variables such as the growth rate (r) or time (t). This is especially useful in scenarios where the variable appears in the exponent, such as in continuous compounding models.

Solving for Variables in Exponential Equations

Solving for variables in exponential equations involves algebraic manipulation and the use of logarithms. In contexts such as determining the interest rate or the initial principal in a continuously compounded growth model, practitioners set up equations based on known data points, isolate the variable of interest through division and logarithmic operations, and then solve for it. This process is fundamentally important in fields such as finance, physics, and natural sciences where exponential modeling is applied.

Exponential Growth Model

The exponential growth model describes situations where the rate of change of a quantity is proportional to the current amount. This is typically represented by the equation A(t) = P e^(rt), where A(t) is the amount at time t, P is the initial principal, r is the growth rate, and t is time. This model is widely used in contexts such as finance, biology, and physics to model continuous growth processes.

Continuous Compounding

Continuous compounding refers to the mathematical limit that compounding interest can reach if it’s calculated and added to the principal an infinite number of times per period. The formula P e^(rt) is utilized to account for this scenario, allowing one to model growth processes where interest is effectively added an infinite number of times over time, leading to a continuously increasing amount.