Problem 33 - Calculus

Question

Explain what is wrong with the statement. At a time when a bank balance \(B \text{dollars},\) which satisfies \(d B / d t=0.08 B-250,\) is \(5000 \text{dollars},\) the balance is going down.

Step-by-Step Solution

Answer

The statement is wrong because the balance is actually increasing, not decreasing.

Step 1: Understand the Differential Equation

The given equation is a first-order linear differential equation of the form \( \frac{dB}{dt} = 0.08B - 250 \). This equation describes the rate at which the balance \( B \) is changing with respect to time \( t \). The rate depends on the current balance \( B \), a rate of increase (or interest rate) of \( 8\% \), and a constant decrease by \( 250 \) units.

Step 2: Substitute the Given Balance

We substitute the given balance \( B = 5000 \) dollars into the differential equation to determine the sign of \( \frac{dB}{dt} \) at this balance.\[\frac{dB}{dt} = 0.08 \times 5000 - 250\]

Step 3: Calculate the Rate of Change

Calculate the rate of change of the balance:Multiply: \( 0.08 \times 5000 = 400 \)Subtract: \( 400 - 250 = 150 \) Thus, \( \frac{dB}{dt} = 150 \).

Step 4: Analyze the Sign of the Rate of Change

The sign of \( \frac{dB}{dt} = 150 \) is positive, which means the balance \( B \) is increasing when \( B = 5000 \) dollars.

Step 5: Conclusion about the Statement

Since the derivative \( \frac{dB}{dt} = 150 \) is positive at \( B = 5000 \) dollars, the statement claiming the balance is going down is incorrect. In reality, the bank balance is increasing.

Key Concepts

Rate of Change

The rate of change in a differential equation gives us the speed and direction of how a variable, like a bank balance, changes over time. It is usually represented by the derivative, in this case, \( \frac{dB}{dt} \). This derivative tells us how fast the balance \( B \) is changing with respect to time \( t \). If the rate of change is positive, the quantity or balance is increasing. If negative, it's decreasing. In our problem, when we calculated \( \frac{dB}{dt} \) and found it to be 150, a positive value, it indicated that the bank balance was in fact growing, contradicting the statement that it was decreasing.

Linear Differential Equation

A linear differential equation is a type of equation that involves a derivative of a function and the function itself. It's called 'linear' because the highest power of the function and its derivatives is one.
In this problem, the equation \( \frac{dB}{dt} = 0.08B - 250 \) is a first-order linear differential equation. This means that the equation involves the first derivative of \( B \), and not higher derivatives like second or third. The structure \( 0.08B - 250 \) shows a relationship where part of the change in \( B \) (our bank balance) is proportional to its current value \( B \) (that's the term \( 0.08B \)), making it a linear relationship. The \(-250\) is a constant that affects this rate either by increasing or decreasing it.

Interest Rate

Interest rate in the context of differential equations often represents the percentage increase of a quantity over time. Here, our equation features an interest component at \( 0.08 \) or 8%. This means for every dollar in the bank, 8% is gained back as interest over the specified time period.
Interest rates act as multipliers; they influence how much the original quantity grows. On its own, a higher interest rate would naturally lead to more rapid growth.
However, in our problem, the growth due to interest is counterbalanced by the deduction \(-250\). The dynamics between the interest growth from \( 0.08B \) and the deduction compound to tell us if our balance is increasing or decreasing. In this case, despite the constant deduction, the positive result from \( 0.08 \times 5000 - 250 \) shows the balance actually increasing due to the interest rate's impact.