Question

Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75\(\%\) compounded continuously.

Short answer

The investment will triple in value after approximately \(\frac{\ln(3)}{0.0575}\) years.

Step-by-step solution

Identify Known Variables

From the problem, we know that \(A = 3P\), effort is made to triple the principal amount, and the rate is given as \(r = 0.0575\) (converted from percent to decimal).

Substitute into Formula

Plugging into the formula we get \(3P = Pe^{0.0575t}\). Since both sides of the equation have P, we can divide both sides by P to give us \(3 = e^{0.0575t}\).

Solve for Time

In order to solve for \(t\), we will need to isolate \(t\). As the equation involves the exponential, we should use the natural logarithm. Apply natural logarithm on both sides to get \(\ln(3) = 0.0575t\).

Calculate Time

Rearrange the equation by dividing both sides by 0.0575 to solve for time \(t = \frac{\ln(3)}{0.0575}\). When the calculation is done, we get a number.

Interpret Result

The result represents the number of years it would take for the initial investment to triple, given an interest rate of 5.75\%. The unit of time is predicated by the way we expressed our rate, hence our result is in years.