The coefficient 7,000 represents the initial amount of the loan before any payments are made.
Supplemental Knowledge
Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. They are commonly used to model situations where growth or decay happens at a constant percentage rate over time. The general form of an exponential function is \(A \cdot B^ { kt} \), where:
- \(A\) represents the initial amount or starting value.
- \(B\) is the base, representing the growth (if greater than 1) or decay (if between 0 and 1) factor.
- \(k\) is a constant that adjusts the rate of growth or decay.
- \(t\) represents time.
In the expression given, 7,000(0.90)^12t, we can break it down as follows: - The coefficient 7,000 represents the initial loan amount before any repayments have been made.
- The base 0.90 indicates that Sully's loan amount decreases by 10% each month (since 1 - 0.10 = 0.90).
- The exponent 12t suggests that this decrease happens monthly over t years (since there are 12 months in a year).
Concepts to Actions
Imagine taking out a loan for the purchase of your new car; initially owing $20,000. Each month you make payments that reduce your balance by an estimated percentage, covering both interest payments and principal repayment; for instance if this percentage drops by approximately 5% each month due to these payments and interest adjustments then an exponential decay function similar to Sully's can model this situation effectively.
For instance, if your initial loan was%20,000 and it decreases by about 5% each month, after one year (12 months), your remaining balance can be calculated using an exponential function like \(20,000 \cdot ( 0.95) ^ { 12} \). This helps you understand how much you will owe over time and plan your finances accordingly.
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