Final answer:
Using the formula for continuous compounding, it will take approximately 9.13 years for a $1500 investment to triple in value at a 12% interest rate. The calculation involves setting the final amount to $4500, since tripling $1500 gives us $4500, and solving for time using the given interest rate.
Step-by-step explanation:
To find how long it will take for $1500 to triple in value with continuous compounding at a 12% interest rate, we will use the formula A = Pert, where A is the final amount, P is the principal amount, r is the interest rate, t is the time in years, and e is the base of the natural logarithm.
First, we set A to triple the principal amount, which gives us $4500 (since 3 times $1500 is $4500). Our principal, P, is $1500. The annual interest rate, r, is 12% or 0.12 in decimal form. We are solving for t, the time it takes to triple the investment.
The formula becomes: $4500 = $1500 * e(0.12t). To solve for t, we first divide both sides by $1500, which gives us 3 = e(0.12t). We then take the natural logarithm of both sides to get ln(3) = 0.12t.
To isolate t, we divide both sides by 0.12, giving us t = ln(3) / 0.12. After evaluating ln(3) / 0.12, we find out that t ≈ 9.13 years. Therefore, it will take approximately 9.13 years for the $1500 to triple in value when compounded continuously at a 12% interest rate.