To determine the appropriate range for the value of the car after 7 years, we should analyze the options in relation to the value function given:
The value of the car after \( t \) years is given by the equation:
\[ v(t) = 40,000(0.89)^t \]
We need to find \( v(7) \):
\[ v(7) = 40,000(0.89)^7 \]
Now, let's calculate \( (0.89)^7 \):
\[ (0.89)^7 \approx 0.5132 \]
Now multiply this by 40,000:
\[ v(7) \approx 40,000 \times 0.5132 \approx 20528 \]
So, after 7 years, the car will be worth approximately $20,528.
Now let's evaluate the options:
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Option #1: \( 0 < t \leq 40,000 \) - This doesn't make sense because \( t \) represents years, not a value, and it suggests a range for values rather than time.
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Option #2: \( 0 < v \leq 40,000 \) - This option suggests that the value of the car is in a range that is not entirely correct since the car depreciates and will be worth less than $40,000 after any period following the purchase.
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Option #3: \( 0 < t \leq 7 \) - This option correctly restricts \( t \) to the time frame of interest (0 to 7 years).
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Option #4: \( 0 < v \leq 7 \) - This option does not make sense as it incorrectly represents \( v \) which is a monetary value, and it should not be compared to years.
The appropriate choice among the given options, focusing on the correct context (time), would be:
Option #3: \( 0 < t \leq 7 \).