To determine how much the car will be worth in 7 years, we can analyze the given function for the value of the car after \( t \) years:
\[ v(t) = 40,000(0.89)^t \]
We are interested in the value of the car after 7 years, so we will calculate \( v(7) \):
\[ v(7) = 40,000(0.89)^7 \]
Now, we need to calculate \( (0.89)^7 \):
\[ (0.89)^7 \approx 0.513 \] (approximately)
Now plug this back into the equation:
\[ v(7) \approx 40,000 \times 0.513 \approx 20,520 \]
Now, we know that the value \( v \) of the car after 7 years will be approximately $20,520.
Now looking at the options:
-
Option #1: \( 0<t≤40,000 \) - This is not appropriate because \( t \) represents time (years), not value.
-
Option #2: \( 0<v≤40,000 \) - This option indicates that the value of the car is in the range from greater than 0 to 40,000. However, since the value of the car will be much less than 40,000 after 7 years, this option does not specifically indicate an appropriate range for 7 years.
-
Option #3: \( 0<t≤7 \) - This option is appropriate as it indicates the time \( t \) (in years) that Karen has until the car is paid off.
-
Option #4: \( 0<v≤7 \) - This is not appropriate because the value of the car at year 7 is much greater than 7.
From the above analysis, Option #3 is the most appropriate choice, as it correctly reflects the time range for the car loan (from 0 to 7 years).
Final answer: Option #3