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

Answer 1

To determine the appropriate range for the value of the car after 7 years, we should analyze the options in relation to the value function given:

The value of the car after \( t \) years is given by the equation:

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

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

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

Now, let's calculate \( (0.89)^7 \):

\[
(0.89)^7 \approx 0.5132
\]

Now multiply this by 40,000:

\[
v(7) \approx 40,000 \times 0.5132 \approx 20528
\]

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

Now let's evaluate the options:

1. **Option #1: \( 0 < t \leq 40,000 \)** - This doesn't make sense because \( t \) represents years, not a value, and it suggests a range for values rather than time.

2. **Option #2: \( 0 < v \leq 40,000 \)** - This option suggests that the value of the car is in a range that is not entirely correct since the car depreciates and will be worth less than $40,000 after any period following the purchase.

3. **Option #3: \( 0 < t \leq 7 \)** - This option correctly restricts \( t \) to the time frame of interest (0 to 7 years).

4. **Option #4: \( 0 < v \leq 7 \)** - This option does not make sense as it incorrectly represents \( v \) which is a monetary value, and it should not be compared to years.

The appropriate choice among the given options, focusing on the correct context (time), would be:

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