Find the time it takes for $7,500 to double when invested at an annual interest rate of 4.7%, compounded continuously. Recall that the continuous compound interest formula is [tex]A(t)=A_0 e^{r t}[/tex]. Give your answer to 2 decimal places.
Answer (1)
Substitute the given values into the continuous compounding formula: 15000 = 7500 e 0.047 t .
Divide both sides by 7500: 2 = e 0.047 t .
Take the natural logarithm of both sides: ln ( 2 ) = 0.047 t .
Solve for t : t = 0.047 l n ( 2 ) ≈ 14.75 years. The time it takes for the investment to double is 14.75 .
Explanation
Understanding the Problem We are given an initial investment of $7,500 that earns interest at an annual rate of 4.7%, compounded continuously. Our goal is to find the time it takes for the investment to double. The formula for continuous compounding is given by A ( t ) = A 0 e r t , where A ( t ) is the amount at time t , A 0 is the initial amount, r is the interest rate, and t is the time in years.
Setting up the Equation We want to find the time t when the investment doubles, so A ( t ) = 2 A 0 = 2 ( $7 , 500 ) = $15 , 000 . We are given A 0 = $7 , 500 and r = 4.7% = 0.047 . Substituting these values into the formula, we get: 15000 = 7500 e 0.047 t
Simplifying the Equation Now, we solve for t . First, divide both sides of the equation by 7500: 7500 15000 = e 0.047 t 2 = e 0.047 t
Applying Natural Logarithm Next, take the natural logarithm of both sides: ln ( 2 ) = ln ( e 0.047 t ) ln ( 2 ) = 0.047 t
Isolating t Now, solve for t by dividing both sides by 0.047: t = 0.047 ln ( 2 )
Calculating the Time Using a calculator, we find: t ≈ 0.047 0.693147 ≈ 14.7478 Rounding to two decimal places, we get t ≈ 14.75 years.
Final Answer Therefore, it takes approximately 14.75 years for the investment to double.
Examples
Continuous compounding is a fundamental concept in finance and is used to model various investment scenarios. For example, consider a small business owner who invests $10,000 in a high-yield savings account with a continuously compounded interest rate of 6.2%. By understanding how to calculate the doubling time, the owner can estimate how long it will take for their investment to reach $20,000, providing valuable insights for financial planning and business growth. This calculation helps in making informed decisions about reinvesting profits or expanding operations, ensuring sustainable financial health.
