Question

the amount of money owed on a certain type of loan has two components: the principal balance of the loan and the amount of interest accrued on the loan. for a loan of this type, each payment was $298.00 and there were 60 payments total. part of each payment was applied to the principal balance of the loan, and the rest was applied to the amount of interest accrued on the loan. the amount of the 1st payment that was applied to the principal balance was $220.93. for each additional payment, the amount that was applied to the principal balance was approximately 0.5% greater than the amount that was applied to the principal balance for the previous payment. which function best approximates the amount for the x th payment, in dollars, that was applied to the amount of interest accrued, where ( x leq 60 )? \\( \boldsymbol{\text{(a)}} \\) ( f(x) = 298.00 - 220.93(1.005)^{x - 1} \\) \\( \boldsymbol{\text{(b)}} \\) ( f(x) = 298.00 - 220.93(1.005)^{x} \\) \\( \boldsymbol{\text{(c)}} \\) ( f(x) = (298.00 - 220.93)(0.995)^{x - 1} \\) \\( \boldsymbol{\text{(d)}} \\) ( f(x) = (298.00 - 220.93)(0.995)^{x} \\)

Explanation:

Step1: Define initial interest payment

First payment interest: $298.00 - 220.93 = 77.07$

Step2: Model growth of principal payment

Principal payment grows by 0.5%: $220.93(1.005)^{x-1}$

Step3: Derive interest payment function

Interest = Total payment - Principal payment:
$f(x) = 298.00 - 220.93(1.005)^{x-1}$

Answer:

A. $f(x) = 298.00 - 220.93(1.005)^{x-1}$