Final answer:
To calculate the time needed for an initial investment of $4,380 to grow to $7,700 at a 5% annual interest rate, the compound interest formula is applied and logarithms are used to solve for the time, yielding approximately 11.9 years, which rounds up to option a. 12 years.
Step-by-step explanation:
To determine how many years you must wait for your $4,380 investment to grow to $7,700 with an interest rate of 5 percent per year, we can use the compound interest formula:
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 in years.
Assuming the interest is compounded annually (n=1), we have:
$7,700 = $4,380(1 + 0.05)t
Dividing both sides by $4,380 gives us:
1.7575 = (1.05)t
Using logarithms to solve for t, we get:
log(1.7575) = t * log(1.05)
t = log(1.7575) / log(1.05) ≈ 11.9
Since you cannot wait for a fraction of a year, you'll have to wait 12 full years for the investment to grow to $7,700. Therefore, the correct answer is (a) 12.