What initial investment at 7.5% compounded continuously for 9 years will accumulate to $60,000?
P e^(.075*9) = 60000
$30,549.39
To find the initial investment needed to accumulate to $60,000 with continuous compounding, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A = the final amount ($60,000)
P = the principal (initial investment)
e = Euler's number, approximately 2.71828
r = interest rate (7.5% or 0.075 as a decimal)
t = time period (9 years)
We can rearrange the formula to solve for the principal (P):
P = A / (e^(rt))
Now, let's substitute the values into the formula:
P = 60000 / (2.71828^(0.075*9))
Using a calculator, we can evaluate this expression:
P ≈ 60000 / 2.71828^(0.675)
P ≈ 60000 / 1.96596
P ≈ 30539.73
Therefore, an initial investment of approximately $30,539.73 is needed to accumulate to $60,000 at a continuous compound interest rate of 7.5% over 9 years.