Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Elmer takes out a $$\$ 100,000$$ loan for a house. He pays money back at a rate of $$\$ 12,000$$ per year. The bank charges him interest at a rate of \(8.5 \%\) per year compounded continuously. Make a continuous model of his economic situation. Write a differential equation whose solution is \(B(t)\), the balance he owes the bank at time \(t\).

Step-by-Step Solution

Answer

The differential equation modeling Elmer's economic situation is \(dB/dt = 0.085B - 12000\).

Step 1: Identify Principal Loan and Repayment Mount

The initial loan amount is $100,000 and Elmer reimburses the bank at a rate of $12,000 a year.

Step 2: Understand Continuous Compounding Interest

The money owed increases at a rate proportional to the amount owed, at a yearly rate of 8.5% or 0.085. This is expressed in mathematical terms as an exponential even with base e, calculated with the formula \(A = Pe^{rt}\), 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 interest rate (decimal), t is the time the money is invested for in years.

Step 3: Formulate the Differential Equation

The yearly rate of change of the balance, \(B(t)\), he owes is the difference between the interest accrued and the repayments made. Therefore, the differential equation for the balance is \(dB/dt = 0.085B - 12000\). This specifies how the balance grows continually from interest and decreases due to repayments.

Key Concepts

Differential Equations

Understanding differential equations is crucial when modeling various real-world phenomena, including finance. At its core, a differential equation represents a relationship between a function and its derivatives. In the context of compounding interest, it captures the ever-changing state of a loan balance over time, accounting for both the interest accrued and the repayments made.

A simple yet powerful example of such an equation is the one given in the exercise: \( \frac{dB}{dt} = 0.085B - 12000 \). In this equation, \( B(t) \) represents Elmer's balance over time, \( t \), and \( \frac{dB}{dt} \) displays how this balance changes at any given moment. Specifically, the equation tells us that the rate of change of the balance (\( \frac{dB}{dt} \)) is the interest accrued at a continuous rate of 8.5% minus the consistent yearly repayment. By solving this equation, one determines the function \( B(t) \) that describes Elmer's economic situation over time.

In layman's terms, think of it like a snapshot of a bank account's ups and downs, with money flowing in and out, captured in a mathematical formula. It's a powerful tool in understanding many financial situations beyond just loan payments.

Exponential Growth

At the heart of continuous compounding lies the principle of exponential growth, which mathematically expresses how quantities grow faster and faster as they continually accumulate. This type of growth is characterized by the fact that the rate of increase is proportional to the current amount. This concept is visually represented by an ever-steepening curve when graphed over time.

The formula \(A = Pe^{rt}\) epitomizes exponential growth in finance, where \(A\) is the amount of money after a certain time, \(P\) is the principal, \(r\) the interest rate, and \(t\) the time in years. It differs from simple interest, where the amount earned is always based on the original principal, by considering the additional interest accrued on the interest that was previously earned, leading to far more significant growth over time.

For Elmer, this means that with each passing year, a larger amount of interest accrues on his loan balance because not only is the original sum gaining interest, but so is the interest from all previous periods. It is a concept that has profound implications in a multitude of disciplines from biology, where it describes population growth, to physics, and of course, finance.

Compound Interest

Elmer's loan illustrates a real-world application of compound interest. Unlike simple interest, which is calculated only on the principal amount, compound interest calculates interest on the principal and also on all accumulated interest. This can happen on an annual, monthly, daily, or, as in Elmer's case, continuous basis.

When it comes to continuous compounding, we use the natural exponential function (e) to model the growth. The formula \(A = Pe^{rt}\), explained earlier, demonstrates how the balance grows when interest is compounded continuously. The 'e' in the formula stands for Euler's number, an irrational and transcendental number that approximately equals 2.71828. It's an essential constant in mathematics that naturally arises when describing growth processes.

In financial terms, this means the interest is being added to the balance at an infinite number of times instantaneously, or theoretically at every moment. Continuous compounding represents the upper limit on how compound interest can accelerate the growth of money. For those managing loans or investments, understanding this concept of compound interest is essential for accurate financial planning and forecasting future balances or returns.