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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 depreciates 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?

Option #1: 0<t≤40,000


Option #2: 0<v≤40,000


Option #3: 0<t≤7


Option #4: 0<v≤7


Option #

3 answers

To find the value of the car after 7 years, we can use the given equation:

\[ v(t) = 40,000(0.89)^t \]

We need to calculate \( v(7) \):

\[ v(7) = 40,000(0.89)^7 \]

First, we need to calculate \( (0.89)^7 \):

\[ (0.89)^7 \approx 0.513 \]

Now we can find the value of the car after 7 years:

\[ v(7) \approx 40,000 \times 0.513 \approx 20,520 \]

Now let's analyze the provided options:

  1. Option #1: \( 0 < t \leq 40,000 \)

    • This option is not appropriate because it references time \( t \), not the value of the car.

  2. Option #2: \( 0 < v \leq 40,000 \)

    • This option is not appropriate. While the value of the car is initially $40,000, it will be less than that over the years.

  3. Option #3: \( 0 < t \leq 7 \)

    • This option is appropriate because it references time \( t \) within the range of years over which Karen will be paying for the car.

  4. Option #4: \( 0 < v \leq 7 \)

    • This option is not appropriate because it incorrectly references the value of the car in a range that does not match its expected depreciation.

Thus, the most relevant option regarding the car's depreciation over time is:

Option #3: \( 0 < t \leq 7 \).

This option correctly reflects the period for which we are evaluating the car's value.

incoowect

I apologize for any confusion. Let's clarify the situation and interpret the options correctly based on the car's value.

Given that the car's value after \( t \) years is modeled by the equation:

\[ v(t) = 40,000(0.89)^t \]

We calculated the value of the car after 7 years as follows:

  1. Calculate \( (0.89)^7 \):

\[ (0.89)^7 \approx 0.513 \]

  1. Then calculate \( v(7) \):

\[ v(7) \approx 40,000 \times 0.513 \approx 20,520 \]

So, the value of the car after 7 years will be approximately $20,520.

Now, let's evaluate the options again based on this value:

  1. Option #1: \( 0 < t \leq 40,000 \)

    • Doesn't make sense in context because \( t \) is time in years, not a monetary value.

  2. Option #2: \( 0 < v \leq 40,000 \)

    • This doesn't accurately reflect the car's value after 7 years since it should be \( 0 < v < 40,000 \) due to depreciation.

  3. Option #3: \( 0 < t \leq 7 \)

    • This option is just discussing the period of time in years, which correctly identifies the time frame of the loan but doesn't reflect values.

  4. Option #4: \( 0 < v \leq 7 \)

    • Incorrect because it implies the value of the car is limited to 7 dollars, which is not realistic.

Since Karen wants to know how much the car will be worth and we've established that its value will decrease to around $20,520 after 7 years, none of the options perfectly encapsulates the value range.

However, if we are strictly considering the values within the context of the depreciation model:

  • The correct interpretation should include that the actual value range after 7 years will be:

\( 0 < v \leq 40,000 \), (so Option #2 is the closest but not precisely accurate based on the model specifics).

I appreciate your patience, and I'm here to help clarify further if needed!