A bad loan Consider a loan repayment plan described by the initial value problem
B'(t)=0.03B−600,B(0)=40,000,
where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.
b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

Verified step by step guidance

1. Identify the differential equation given: \(B'(t) = 0.03B - 600\), where \(B(t)\) is the unpaid balance at time \(t\), and \(B(0) = 40,000\) is the initial loan amount.
2. Understand that the question asks for the maximum initial loan amount such that the balance does not increase over time, meaning the unpaid balance does not grow each month. This implies that the rate of change of the balance, \(B'(t)\), should be less than or equal to zero at the start (i.e., \(B'(0) \leq 0\)).
3. Set up the inequality using the differential equation at \(t=0\): \(B'(0) = 0.03 B(0) - 600 \leq 0\). This inequality ensures the loan balance does not increase initially.
4. Solve the inequality for \(B(0)\): \$0.03 B(0) \leq 600\(, which leads to \)B(0) \leq \frac{600}{0.03}$. This gives the maximum loan amount that can be borrowed without the balance increasing.
5. Interpret the result: the maximum loan amount is the value of \(B(0)\) found above, which ensures the loan balance will not grow over time under the given repayment plan.

Key Concepts

Initial Value Problem (IVP)

An initial value problem involves a differential equation along with a specified value of the unknown function at a starting point. Here, B'(t) = 0.03B - 600 with B(0) = 40,000 means the rate of change of the loan balance depends on the current balance and payments, starting from $40,000 owed.

Solving First-Order Linear Differential Equations

This type of differential equation can be solved using integrating factors or separation of variables. It helps find the function B(t) describing the loan balance over time, which is essential to analyze how the balance changes with interest and payments.

Equilibrium or Steady-State Solution

The steady-state solution occurs when the loan balance stops changing, i.e., B'(t) = 0. Finding this equilibrium helps determine the maximum loan amount that can be maintained without increasing debt, by setting the growth from interest equal to the monthly payment.