Question

Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25\(\%\) compounded continuously.

Short answer

The time required for an investment to double in value under a 6.25% continuously compounded interest rate is approximately 11.1 years.

Step-by-step solution

Set Up the Equation

The formula for calculating continuous compound interest is \(A = Pe^{rt}\). In this case, the aim is to double the principal, implying \(A = 2P\), hence formulating the equation as \(2P = Pe^{6.25t / 100}\). After canceling out \(P\) on both sides, the new equation to solve becomes \(2 = e^{6.25t / 100}\).

Take Natural Logarithm

In order to eliminate the exponent, take the natural log on both sides of this equation. Employing the properties of logarithms, the equation becomes \(ln(2) = ln(e^{6.25t / 100}) = (6.25t / 100) * ln(e)\). Given that the natural logarithm of \(e\) is 1, the equation simplifies to \(ln(2) = 6.25t / 100\).

Solve for Time

Start solving the equation for \(t\). Multiply both sides by \(100 / 6.25\) to isolate \(t\). This yields \(t = 100 * ln(2) / 6.25\). Using a calculator or a suitable tool to compute the natural log of 2, find that \(t \approx 11.1\) years.