Final answer:
To calculate the principal P needed to grow to $3400 in 6 years with continuous compounding at a 5% rate, the formula A = Pert is used. After solving, the initial deposit needed is approximately $2519, which is not included among the provided answer choices, indicating a potential error in the question or answer options.
Step-by-step explanation:
To find out how much money needs to be deposited to obtain $3400 in 6 years with an interest rate of 5% per year compounded continuously, we use the continuous compounding formula:
A = Pert
Where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
t is the time in years.
e is the base of the natural logarithm (approximately equal to 2.71828).
We have A = $3400, r = 5% or 0.05, and t = 6 years. We need to solve for P:
$3400 = P * e(0.05 * 6)
First, calculate e(0.05 * 6):
e(0.05 * 6) = e0.3 ≈ 1.34986
Now divide both sides by 1.34986 to solve for P:
P = $3400 / 1.34986 ≈ $2518.76
Since the question provides options, the closest option to $2518.76 is B. $3000, which is incorrect. Therefore, the original question may potentially have a mistake, or the provided options may not include the correct answer. The actual deposit needed can be rounded to approximately $2519.