To calculate the amount of money that needs to be deposited now to obtain $3,400 in 6 years with continuous compounding at an interest rate of 6.5% per year, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the future amount ($3,400)
P = the principal amount (initial deposit)
e = the base of the natural logarithm (approximately 2.71828)
r = the interest rate (6.5% or 0.065)
t = the time period in years (6 years)
We can rearrange the formula to solve for P:
P = A / e^(rt)
Substituting the given values:
P = 3400 / e^(0.065 * 6)
Using a calculator, we can evaluate the exponential term:
P = 3400 / e^(0.39)
P ≈ 3400 / 1.476
P ≈ 2302.59
Therefore, approximately $2,302.59 needs to be deposited now into the account to obtain $3,400 in 6 years with continuous compounding at an interest rate of 6.5% per year.