Final answer:
To find out how long it will take for a $5000 deposit to grow to $10,000 with an 8% annual interest rate compounded quarterly, you can use the compound interest formula. After applying logarithms to isolate the variable for time, it's determined that it will take just over 9 years to reach the goal.
Step-by-step explanation:
To determine how long it will take for an account balance to reach $10,000 when $5000 is deposited into an account that pays an annual interest rate of 8% compounded quarterly, one can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
We are looking for t when A is $10,000, P is $5,000, r is 0.08 (8% expressed as a decimal), and n is 4 (since the interest is compounded quarterly).
Following the formula, we have:
10,000 = 5,000(1 + 0.08/4)^(4t)
To isolate t, we'll need to use logarithms:
2 = (1 + 0.02)^(4t)
log(2) = 4t * log(1.02)
t = log(2) / (4 * log(1.02))
After calculating the above, we find that t is approximately 9.006, meaning it will take a little over 9 years for the account balance to reach $10,000.