Question

select the correct answer from each drop - down menu.
the rivera and patel families each bought a car at the same time.
the resale values, in dollars, of each car are modeled by these functions, where x is the number of years that the family has owned the car.

rivera family car	patel family car
	0	21,989
	2	17,811
	4	14,427
	6	11,686

the $\square$ 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 $\square$ the resale value of the patel family’s car.

Explanation:

Step1: Find initial resale value (x=0)

For Rivera: \( f(0) = 21249(0.88)^0 = 21249 \)
For Patel: \( g(0) = 21989 \)
Compare: \( 21989 > 21249 \), so Patel has greater initial value.

Step2: Calculate average rate of decrease (first 6 years)

Average rate = \( \frac{\text{Initial} - \text{Final}}{\text{Time}} \)
Rivera: Initial = 21249, Final = \( f(6) = 21249(0.88)^6 \approx 21249(0.4644) \approx 9860 \)
Rate: \( \frac{21249 - 9860}{6} \approx \frac{11389}{6} \approx 1898.17 \)

Patel: Initial = 21989, Final = 11686
Rate: \( \frac{21989 - 11686}{6} = \frac{10303}{6} \approx 1717.17 \)

Compare rates: \( 1898.17 > 1717.17 \), so Rivera's rate is greater (faster decrease).

Answer:

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 greater than the resale value of the Patel family’s car.