Final answer:
The time required for an investment to grow to $8300 at an interest rate of 7.5% per year, compounded quarterly, is approximately 6.72 years.
Step-by-step explanation:
To find the time required for an investment to grow, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount
r = the annual interest rate
n = the number of times the interest is compounded per year
t = the time in years
In this case, we have:
A = $8300
P = $5000
r = 7.5% = 0.075
n = 4 (quarterly compounds)
Now, we can solve for t
$8300 = $5000(1 + 0.075/4)^(4t)
To find the value of t, we can take the logarithm of both sides:
log($8300) = log($5000(1 + 0.075/4)^(4t)
log($8300) = log($5000) + log((1 + 0.075/4)^(4t)
We can simplify the equation:
log($8300) = log($5000) + 4tlog(1 + 0.075/4)
Now, we can solve for t. Using a calculator:
t = (log($8300) - log($5000))/(4log(1 + 0.075/4))
t ≈ 6.72 years (rounded to two decimal places)