Now, using the best fit function for the data, [tex]y=-465.4569 x+17,267.6[/tex], perform extrapolation to estimate the principal balance of the loan after 15, 20, and 30 payments. Type the correct answer in each box. Round your answers to the nearest cent.
Answer (2)
By substituting specific payment numbers into the equation y = − 465.4569 x + 17267.6 , we can estimate the loan balance after 15, 20, and 30 payments as $10285.15, $7958.46, and $3303.89 respectively. Each calculated balance rounds to the nearest cent for clarity. These values help in understanding how loan payments affect the remaining balance over time.
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Substitute x = 15 into the equation y = − 465.4569 x + 17267.6 to find the balance after 15 payments: y 1 = − 465.4569 ( 15 ) + 17267.6 = 10285.75 .
Substitute x = 20 into the equation y = − 465.4569 x + 17267.6 to find the balance after 20 payments: y 2 = − 465.4569 ( 20 ) + 17267.6 = 7958.46 .
Substitute x = 30 into the equation y = − 465.4569 x + 17267.6 to find the balance after 30 payments: y 3 = − 465.4569 ( 30 ) + 17267.6 = 3303.89 .
The estimated principal balances after 15, 20, and 30 payments are $10285.75 , $7958.46 , and $3303.89 , respectively.
Explanation
Understanding the Problem We are given the best fit function y = − 465.4569 x + 17267.6 , where y represents the principal balance of a loan and x represents the number of payments. We need to estimate the principal balance after 15, 20, and 30 payments.
Calculating Balance After 15 Payments To estimate the principal balance after 15 payments, we substitute x = 15 into the equation: y 1 = − 465.4569 ( 15 ) + 17267.6
Result After 15 Payments Calculating the value: y 1 = − 6981.8535 + 17267.6 = 10285.7465 Rounding to the nearest cent, we get y 1 = 10285.75 .
Calculating Balance After 20 Payments To estimate the principal balance after 20 payments, we substitute x = 20 into the equation: y 2 = − 465.4569 ( 20 ) + 17267.6
Result After 20 Payments Calculating the value: y 2 = − 9309.138 + 17267.6 = 7958.462 Rounding to the nearest cent, we get y 2 = 7958.46 .
Calculating Balance After 30 Payments To estimate the principal balance after 30 payments, we substitute x = 30 into the equation: y 3 = − 465.4569 ( 30 ) + 17267.6
Result After 30 Payments Calculating the value: y 3 = − 13963.707 + 17267.6 = 3303.893 Rounding to the nearest cent, we get y 3 = 3303.89 .
Final Answer Therefore, the estimated principal balances after 15, 20, and 30 payments are $10285.75 , $7958.46 , and $3303.89 , respectively.
Examples
Understanding loan balances is crucial in personal finance. For instance, if you're planning to refinance your loan or make extra payments, knowing the estimated balance at different points in time helps you make informed decisions. This calculation is also useful in business for forecasting financial liabilities and planning investments. By using a best-fit function, you can predict future balances and adjust your financial strategies accordingly, ensuring better financial health and stability.
