Problem 37 - Calculus with Concepts in Calculus

Question

A real estate firm can borrow money at \(5 \%\) interest per year and can lend the money out. If the amount of money it can lend is inversely proportional to the square of the interest rate at which it lends, what interest rate would maximize the firm's profit per year? (Hint: Let \(x\) be the loan interest rate. Notice that the profit is the product of the amount borrowed by the firm and the difference between the interest rates at which it lends and borrows.)

Step-by-Step Solution

Answer

The interest rate that maximizes the firm's profit is 10%.

Step 1: Define the Variables

Let \( x \) be the loan interest rate and \( b \) be the amount the firm can borrow. From the problem, the amount \( A \) that the firm can lend is inversely proportional to the square of \( x \). Therefore, we write \( A = \frac{k}{x^2} \) where \( k \) is a constant.

Step 2: Express the Profit Function

The profit \( P \) of the firm per year is the product of the amount it can lend and the difference in interest rates, i.e., \( P = (x - 0.05) \cdot \frac{k}{x^2} = \frac{k(x - 0.05)}{x^2} \). This is the function we need to maximize.

Step 3: Find the Derivative of the Profit Function

Differentiate the profit function with respect to \( x \). Using the quotient rule, \( P'(x) = \frac{(x^2 \cdot k) - (k(x - 0.05) \cdot 2x)}{x^4} \). Simplifying, we find \( P'(x) = \frac{k(x^2 - 2x(x - 0.05))}{x^4} = \frac{k(0.1x - x^2)}{x^4} \).

Step 4: Set the Derivative to Zero

Set the derivative to zero to find the critical points: \( \frac{k(0.1x - x^2)}{x^4} = 0 \). This simplifies to \( 0.1x - x^2 = 0 \). Factoring out \( x \), we have \( x(0.1 - x) = 0 \). Thus, \( x = 0 \) or \( x = 0.1 \).

Step 5: Analyze the Critical Points

Since \( x = 0 \) is not feasible for a positive interest rate, we consider \( x = 0.1 \). We check that this gives a maximum by considering the second derivative or a sign change test for derivatives around \( x = 0.1 \).

Step 6: Confirm the Maximization

Verify that when \( x = 0.1 \) using changes in sign of \( P'(x) \), the function changes from positive to negative, confirming a local maximum.

Key Concepts

Inversely Proportional

In mathematical terms, when we say that two variables are inversely proportional, it means that as one variable increases, the other decreases, and vice versa. Specifically, the product of these two variables remains constant. In this exercise, the firm can lend an amount that is inversely proportional to the square of the interest rate. This gives us the relationship: \( A = \frac{k}{x^2} \), where \( A \) is the amount lent, \( x \) is the interest rate, and \( k \) is a constant.

What this relationship tells us is that if the interest rate (\( x \)) goes up, the amount \( A \) that the firm can lend must decrease to maintain the same value of \( k \). Conversely, if the interest rate falls, the firm can lend more. This principle is central to understanding how changes in interest rates impact the firm's lending capabilities and ultimately its profits.

Interest Rate

The interest rate is the percentage at which interest is charged or paid. It acts as a crucial variable in financial scenarios, determining the cost of borrowing money or the return on lending it. In this exercise, the firm's financial transactions are influenced heavily by the interest rates at which money is borrowed and lent.

The firm borrows at a rate of 5%, but lends at a rate that is represented by \( x \). The interest rate \( x \) acts as a lever affecting both the profitability through its relation to the borrow rate and the amount that can be lent. Understanding interest rates and manipulating them effectively can lead to optimal financial decisions and profit maximization.

Profit Maximization

Profit maximization is the process of determining the best level at which operations should occur to yield the greatest profit. For the firm, profit is calculated as the difference between the interest earned on money lent and the interest paid on money borrowed, multiplied by the amount of money available to lend. The profit formula derived here is given by \( P = \frac{k(x - 0.05)}{x^2} \).

To maximize profit, the firm must carefully choose its lending interest rate \( x \). By setting the derivative of the profit function equal to zero, the firm finds the critical points where profit could be at a maximum. Evaluating these points reveals the optimal interest rate \( x = 0.1 \) for maximizing profit. This showcases the importance of calculus in making strategic financial decisions.

Quotient Rule

The quotient rule is a fundamental tool in calculus used to differentiate functions that are the ratio of two differentiable functions. If you have a function \( f(x) = \frac{u(x)}{v(x)} \), the derivative is \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{v(x)^2} \).

In the solution to the problem, the profit function \( P(x) = \frac{k(x - 0.05)}{x^2} \) is differentiated using the quotient rule. Applying this rule helps find \( P'(x) \), which then is set to zero to identify critical points. By calculating \( P'(x) \), we can determine the interest rates where profit changes, thereby assisting in finding where the maximum profit occurs. Understanding the quotient rule is essential for tackling optimization problems in calculus.