Questions: How long will it take for an investment to triple, if interest is compounded continuously at 6%? It will take years before the investment triples. (Round to the nearest tenth of a year.)

How long will it take for an investment to triple, if interest is compounded continuously at 6%?

It will take years before the investment triples. (Round to the nearest tenth of a year.)

Transcript text: How long will it take for an investment to triple, if interest is compounded continuously at $6 \%$ ? It will take $\square$ years before the investment triples. (Round to the nearest tenth of a year.)

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the time $T$ it will take for an investment to triple in value with continuous compounding at rate $r\%$.

Step 2: Apply the Continuous Compounding Formula

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate in decimal, and $t$ is the time in years.

Step 3: Set $A = 3P$ and Solve for $t$

Given that the investment triples, we set $A = 3P$, leading to $3 = e^{rt}$. Taking the natural logarithm of both sides gives $\ln(3) = rt$, which simplifies to $t = rac{\ln(3)}{r}$.

Step 4: Convert $r$ from Percentage to Decimal

Since $r$ is given as a percentage, we convert it to a decimal by dividing by 100: $r = rac{r}{100}$.

Step 5: Calculate $T$

Substituting the given rate $r = 6\%$ into the formula, we get $T = rac{\ln(3)}{r/100} = 18.3$ years.