Questions: The amount A of dollars accumulated after t years through an investment of A0 dollars at an interest rate of r, compounded continuously, can be determined from the function A=A0 e^rt. The doubling time of an investment is the time required for an investment to double in value. Find the doubling time (rounded to 3 decimal places) for an investment made at a. 8% interest compounded continuously. years

The amount A of dollars accumulated after t years through an investment of A0 dollars at an interest rate of r, compounded continuously, can be determined from the function A=A0 e^rt. The doubling time of an investment is the time required for an investment to double in value. Find the doubling time (rounded to 3 decimal places) for an investment made at a. 8% interest compounded continuously.
years

Transcript text: The amount $A$ of dollars accumulated after $t$ years through an investment of $A_{0}$ dollars at an interest rate of $r$, compounded continuously, can be determined from the function $A=A_{0} e^{r t}$. The doubling time of an investment is the time required for an investment to double in value. Find the doubling time (rounded to 3 decimal places) for an investment made at a. $8 \%$ interest compounded continuously. $\square$ years

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the time $T$ it takes for an initial investment to double when invested at an annual interest rate $r$, compounded continuously.

Step 2: Apply the Formula for Continuous Compounding

The formula for continuous compounding is $P = P_0e^{rt}$, where $P$ is the future value, $P_0$ is the initial investment, $r$ is the annual interest rate, and $t$ is the time in years.

Step 3: Set $P = 2P_0$ to Find the Doubling Time

Since we want the investment to double, we set $P = 2P_0$, which gives us $2 = e^{rt}$.

Step 4: Solve for $t$

Taking the natural logarithm of both sides gives us $\ln(2) = rt$. Solving for $t$, we get $T = \frac{\ln(2)}{r}$.

Step 5: Calculate the Doubling Time

Substituting $r = 0.08$ into the formula, we get $T = \frac{\ln(2)}{0.08} = 8.664$ years.