Final answer:
To find the time it takes for $3200 to grow to $9000 at an 8% interest rate, we use the compound interest formula with annual compounding. After calculating and solving for the time variable 't', we round to the nearest tenth of a year.
Step-by-step explanation:
To determine how long it will take for an initial investment of $3200 to grow to $9000 at an annual interest rate of 8%, we'll use the formula for compound interest, which is 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.
Assuming the interest is compounded annually (n = 1), our formula simplifies to A = P(1 + r)t.
Now, let's plug in the numbers to solve for t:
9000 = 3200(1 + 0.08)t
Dividing both sides by 3200, we get:
(9000/3200) = (1.08)t
Taking the natural logarithm of both sides:
ln(9000/3200) = t * ln(1.08)
Finally, solve for t:
t = ln(9000/3200) / ln(1.08)
After calculating the above expression, we round t to the nearest tenth of a year to find the answer.