Question

The equation for the amount of money in an account at a specific bank is A = P(1 + r/n)^(nt), where A is the final amount after t years at an annual interest rate of r compounded n times a year, and P is the initial investment. If you deposit $5,000 into an account paying 6% annual interest compounded monthly, how long until there is $8,000 in the account?

a) Approximately 5.92 years
b) Approximately 7.20 years
c) Approximately 8.85 years
d) Approximately 9.60 years

Answer 1

To find out how long it takes for the account to reach $8,000, we need to solve for t in the equation A = P(1 + r/n)^(nt). By substituting the given values into the equation and solving for t, we find that it takes approximately 5.92 years for the account to reach $8,000.

Step-by-step explanation:

To find out how long it takes for the account to reach $8,000, we need to solve for t in the equation A = P(1 + r/n)^(nt). In this case, P = $5,000, A = $8,000, r = 0.06, and n = 12 (since interest is compounded monthly). Substituting these values into the equation and solving for t:

8000 = 5000(1 + 0.06/12)^(12t)

Divide both sides by 5000:

1.6 = (1.005)^12t

Take the natural logarithm of both sides:

ln(1.6) = ln((1.005)^12t)

Use the property of logarithms that allows us to bring the exponent down:

ln(1.6) = 12t * ln(1.005)

Divide both sides by 12 * ln(1.005):

t = ln(1.6) / (12 * ln(1.005))

Using a calculator, we find that t ≈ 5.92 years. Therefore, the answer is approximately 5.92 years (option a).