Final answer:
Using the compound interest formula, Juan will need approximately 7.95 years for his $5000 investment at an annual rate of 8.6%, compounded semiannually, to grow to $7945.
Step-by-step explanation:
Juan needs $7945 for a future project. He can invest $5000 now at an annual rate of 8.6%, compounded semiannually. We will use the compound interest formula to determine how long it will take for his investment to grow to this amount. The formula for compound interest is:
A = P(1 + r/n)(nt)
where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money).
r = the annual interest rate (decimal).
n = the number of times that interest is compounded per year.
t = the time the money is invested for, in years.
In Juan's case:
- P = $5000
- r = 8.6% or 0.086
- n = 2 (since interest is compounded semiannually)
- A = $7945
We will solve for t:
$7945 = $5000(1 + 0.086/2)(2t)
Now, divide both sides by $5000:
1.589 = (1 + 0.043)(2t)
Then, take the natural logarithm of both sides to solve for t:
ln(1.589) = ln((1.043)(2t))
= 2t · ln(1.043)
After calculating the left and right side:
t = ln(1.589) / (2 · ln(1.043))
After performing the calculations, we will round the answer to the nearest hundredth:
t ≈ 7.95 years
It will take approximately 7.95 years for Juan's investment to grow to $7945.