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Karen purchased a car for $40,000 by taking out a loan that will take 7 years to pay off. Given the car depreciated in value by 11% each year, the situation can be modeled by the equation v(t) = $40,000(0.89)^t, where v is the value of the car after t years. If Karen wants to know how much the car will be worth in 7 years, which of the following options would be an appropriate range

3 answers

To find out how much the car will be worth after 7 years, we can use the given depreciation formula:

\[ v(t) = 40000(0.89)^t \]

We need to substitute \(t = 7\) into the equation:

\[ v(7) = 40000(0.89)^7 \]

Now, calculating \(0.89^7\):

\[ 0.89^7 \approx 0.5132 \]

Now, we can calculate \(v(7)\):

\[ v(7) \approx 40000 \times 0.5132 \approx 20528 \]

Thus, after 7 years, the car will be worth approximately $20,528.

So, an appropriate range for the car's worth after 7 years would be around $20,000 to $21,500.

Options
O< t <=7
0< v <= 7
0< t <= 40,000
0< v<=40,000

To analyze the options given:

  1. \(0 < t \leq 7\): This option suggests that \(t\) can take on values from 0 to 7 years. It is a valid range for the time in years since Karen is considering the duration over which she is paying off the loan.

  2. \(0 < v \leq 7\): This option suggests that the value \(v\) is between 0 and 7, which does not make sense in the context since the car's value in dollars is expected to be much higher. This option is not suitable.

  3. \(0 < t \leq 40,000\): This option implies that time \(t\) can be from 0 to 40,000, which is unreasonable since time is measured in years and doesn't stretch to that range. This option is not valid.

  4. \(0 < v \leq 40,000\): This option implies that the value \(v\) of the car can take on any value greater than 0 and up to 40,000. Given that the car starts off with a value of $40,000 and depreciates over time, this range is appropriate because the car will eventually lose value but will not drop below 0.

Based on the calculations we did earlier:

  • The car's worth after 7 years is estimated at approximately $20,528.

The most appropriate choice from the options provided would be:

  • \(0 < v \leq 40,000\), as it reasonably encompasses the car’s depreciation value over time.