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?

Question

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 #

Answer 1

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: $0<t\le 40,000$ - This is not appropriate because $t$ represents time (years), not value.

2. Option #2: $0<v\le 40,000$ - 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: $0<t\le 7$ - This option is appropriate as it indicates the time $t$ (in years) that Karen has until the car is paid off.

4. Option #4: $0<v\le 7$ - 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