To determine how much the car will be worth in 7 years, we need to evaluate the function \( v(t) = 40,000(0.89)^t \) at \( t = 7 \).
First, we calculate \( v(7) \):
\[ v(7) = 40,000(0.89)^7 \]
Calculating \( (0.89)^7 \):
\[ (0.89)^7 \approx 0.513 \quad (\text{approximately}) \]
So we have:
\[ v(7) \approx 40,000 \times 0.513 \approx 20,520 \]
After 7 years, the car will be worth approximately $20,520.
Next, we need to analyze the options given.
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Option #1: \( 0 < t \leq 40,000 \) - This option does not make sense since \( t \) represents time in years, not values.
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Option #2: \( 0 < v \leq 40,000 \) - This suggests that the value can range up to $40,000, which is incorrect because in 7 years the value will be much less than $40,000. Thus, this option is also not appropriate.
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Option #3: \( 0 < t \leq 7 \) - This option correctly shows the range for \( t \), which represents the time in years until the loan is paid off (7 years).
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Option #4: \( 0 < v \leq 7 \) - This does not apply because \( v \) represents the car's value, which is expected to much exceed 7 dollars.
Considering the evaluation of the problem, the most appropriate option that makes sense in this context is:
Option #3: \( 0 < t \leq 7 \).