Money is deposited in an account for which the interest is compounded continuously. If the balance doubles in 6 years, what is the annual percentage rate?
Can someone show me how to set this up? I have to use the equation: M(t)= Ce^(kt). Thanks.
To solve this problem using the equation M(t) = Ce^(kt), where M(t) represents the balance at time t, C represents the initial deposit, k represents the constant interest rate, and e is Euler's number (approximately 2.71828), you can follow these steps:
Step 1: Gather the given information
- The balance doubles in 6 years. This means that M(6) = 2C.
- Note that the equation represents continuous compounding, so the interest rate is compounded continuously over time.
Step 2: Determine the unknowns
- The unknowns in this problem are C and k. We need to find the value of k, which represents the annual percentage rate (APR) in continuous compounding.
Step 3: Set up the equation
- Since the balance doubles in 6 years, we have M(6) = Ce^(6k) = 2C.
Step 4: Solve for k
- Divide both sides of the equation by C to eliminate it: e^(6k) = 2.
- Take the natural logarithm (ln) of both sides of the equation: ln(e^(6k)) = ln(2).
- The natural logarithm of e^(6k) cancels out, leaving us with: 6k = ln(2).
- Now, divide both sides of the equation by 6 to solve for k: k = (1/6)ln(2).
Step 5: Calculate the APR
- To convert k into an annual percentage rate (APR), we need to multiply k by 100: APR = (1/6)ln(2) * 100.
- Plug this value into your calculator to get the result.
So, using the equation M(t) = Ce^(kt), and given that the balance doubles in 6 years, the annual percentage rate (APR) can be determined by calculating (1/6)ln(2) * 100.