Question
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
(1 point)
Option #
Option #1: 0<t≤40,000
Option #2: 0<v≤40,000
Option #3: 0<t≤7
Option #4: 0<v≤7
(1 point)
Option #
Answers
GPT-4o mini
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:
1. **Option #1: \( 0<t≤40,000 \)** - This is not appropriate because \( t \) represents time (years), not value.
2. **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.
3. **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.
4. **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**
\[ v(t) = 40,000(0.89)^t \]
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:
1. **Option #1: \( 0<t≤40,000 \)** - This is not appropriate because \( t \) represents time (years), not value.
2. **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.
3. **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.
4. **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**