Answer: To solve this problem, we can use the formula for calculating the future value of an investment:
FV = PV * (1 + r/n)^(nt)
where:
FV is the future value of the investment
PV is the present value of the investment (the initial amount Louise has to invest)
r is the annual interest rate (4.28%)
n is the number of times the interest is compounded in a year
t is the number of years the investment is made
Since the interest is compounded monthly, we need to convert the annual interest rate to a monthly rate:
r/12 = 4.28% / 12 = 0.3566666666666667%
Now, we can use the formula to calculate the future value of Louise's investment:
FV = $20,000 * (1 + 0.3566666666666667%)^(12t)
We can use trial and error or iterative methods to find the value of t that satisfies the condition: FV = $30,000.
A quick approximation is to use the formula for simple interest:
FV = PV * (1 + r * t) = $20,000 * (1 + 0.0428 * t)
t = ($30,000 - $20,000) / ($20,000 * 0.0428) = 2.857142857142857 years, or approximately 2.9 years.
This is an approximation, but it should be close to the actual answer. To get a more accurate answer, we can use an iterative method, such as Newton-Raphson or bisection, to solve for t in the formula above.
Explanation: