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 #__

Answer 1

the answer is option 2

Answer 2

To determine how much the car will be worth in 7 years, we need to evaluate the function \( v(t) = 40,000(0.89)^t \) at \( t = 7 \).

First, we calculate \( v(7) \):

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

Calculating \( (0.89)^7 \):

\[
(0.89)^7 \approx 0.513 \quad (\text{approximately})
\]

So we have:

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

After 7 years, the car will be worth approximately $20,520.

Next, we need to analyze the options given.

- **Option #1: \( 0 < t \leq 40,000 \)** - This option does not make sense since \( t \) represents time in years, not values.

- **Option #2: \( 0 < v \leq 40,000 \)** - This suggests that the value can range up to $40,000, which is incorrect because in 7 years the value will be much less than $40,000. Thus, this option is also not appropriate.

- **Option #3: \( 0 < t \leq 7 \)** - This option correctly shows the range for \( t \), which represents the time in years until the loan is paid off (7 years).

- **Option #4: \( 0 < v \leq 7 \)** - This does not apply because \( v \) represents the car's value, which is expected to much exceed 7 dollars.

Considering the evaluation of the problem, the most appropriate option that makes sense in this context is:

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