If the prevailing interest rate is 3.5%/year compounded continuously, find the future value of this income stream. (Round your answer to two decimal places.)

Question

The newly opened Mario's Trattoria is expected to produce a continuous income stream at the rate of R(t) = 120,000 dollars/year for the next 4 years.

Answer 1

Terms:

1. Continuous Compounding: Interest is calculated and added to the account balance continuously.
2. Future Value = FV = The amount of money an investment will grow to over time with continuous compounding.

FV = ∫(from 0 to T) ​R(t)⋅e^r(T-t)dt

Where:

1. T is the total time period (4 years)
2. R(t) is the income stream rate (120,000 dollars/year)
3. r is the continuous compounding interest rate (3.5% or 0.035)

Steps to Solve:

1. Set Up the Integral: FV = ∫[from 0 to 4] ​120,000⋅e^0.035(4−t) dt
2. Evaluate and simplify the Exponent: FV=120,000 ∫[from 0 to 4] e^(0.14−0.035t) dt
3. Change of Variables:

Let u = 0.14 − 0.035t

du=−0.035 dt

Adjust the limits of integration:

1. When t=0, u=0.14
2. When t=4, u=0

FV=120,000 ∫[from 0 to 0.140] e^u⋅(du/−0.035​)

1. Integrate, with your final result should be: FV = −120K/0.035*​(1−e^0.14)
2. Calculate the Values:

e^0.14 ≈ 1.15027

FV = −120,000/0.035 * (1−1.15027)

FV ≈ 515,211.43 - Final answer.