Let's break down the provided information step-by-step to correctly fill in the blanks about the resale values of the Rivera and Patel family cars.
1. Initial Resale Values:
- The Rivera family's car can be described by the function [tex]\( f(x) = 21,249 (0.88)^x \)[/tex] for the resale value over the years.
- The Patel family's car has given the resale values at certain years:
- At [tex]\( x = 0 \)[/tex] (initial), [tex]\( g(0) = 21,989 \)[/tex]
- Other resale values are also given but are not required for this particular blank.
Therefore, comparing the initial resale values:
- Rivera family's initial resale value: 21,249
- Patel family's initial resale value: 21,989
Clearly, the Patel family’s car had the greater initial resale value.
2. Average Rate of Decrease in Resale Value:
- Rivera family's car has resale values modeled by [tex]\( f(x) \)[/tex]. To determine the average rate of decrease over 6 years:
- Calculate the resale value at [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 21,249 \)[/tex]
- Calculate the resale value at [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 21,249 \times (0.88)^6 \)[/tex]
(According to the earlier provided answer, [tex]\( f(6) \approx 9,836.12444192782 \)[/tex])
- The average rate of decrease over 6 years is then:
[tex]\[
\text{Average decrease} = \frac{f(0) - f(6)}{6} = \frac{21,249 - 9,836.12444192782}{6} \approx 1,896.81
\][/tex]
Thus, putting this all together:
- The Patel family's car had the greater initial resale value.
- During the first six years, the resale value of the Rivera family's car decreases at an average rate of 1,896.81 per year.
