Problem 72 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

FUTURE VALUE OF AN INCOME STREAM Money is transferred continuously into an account at the rate of \(5,000 e^{0.015 t}\) dollars per year at time \(t\) (years). The account earns interest at the annual rate of \(5 \%\) compounded continuously. How much will be in the account at the end of 3 years?

Step-by-Step Solution

Answer

The amount in the account at the end of 3 years will be approximately $16,529.76.

Step 1: Identify the given data

The rate of money transfer into the account is given by the function: \( \text{Rate}(t) = 5,000 e^{0.015t} \) The annual interest rate (continuously compounded) is: \( r = 0.05 \)

Step 2: Set up the integral for future value

To find the future value, use the formula for continuous income streams with continuous compounding interest: \[ FV = \text{e}^{rt} \times \text{Rate}(t) \times dt \ FV = \int_0^3 5000 e^{0.015t} e^{0.05(3 - t)} dt \]

Step 3: Simplify the integrand

Combine the exponential terms inside the integral: \[ FV = 5000 \int_0^3 e^{0.015t} e^{0.15 - 0.05t} dt = 5000 \int_0^3 e^{0.15 + (0.015 - 0.05)t} dt \] \[ FV = 5000 \int_0^3 e^{0.15 - 0.035t} dt \]

Step 4: Integrate the function

Integrate \(e^{0.15 - 0.035t}\) with respect to \(t\): \[ 5000 \int_0^3 e^{0.15 - 0.035t} dt = \frac{5000}{-0.035} [e^{0.15 - 0.035t}]_0^3 \] \[ = -142857.14 [e^{0.15 - 0.105} - e^{0.15}] \]

Step 5: Evaluate the expression

Substitute the limits 0 and 3 into the evaluated function: \[ = -142857.14 [e^{0.15 - 0.105} - e^{0.15}] \] \[ = -142857.14 [e^{0.045} - e^{0.15}] \] \[ = -142857.14 [1.04603 - 1.16183] \] \[ = -142857.14 (-0.1158) \] \[ = 16529.76 \]

Key Concepts

continuous compounding interest

Continuous compounding interest is a powerful concept in finance that allows money to grow at a constant rate over time. Unlike simple interest that is calculated periodically, continuous compounding assumes interest is reinvested instantly. This means the interest itself earns interest continuously. The formula for continuous compounding interest is given by:

<[...] Where: \ \[ FV = Future value : estprinciple accrued.}Initialization Mode! \ : : ? \Rate_RATE }\ V visualization\] ; Here we use this con'formula :
< DIR : =