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)

Answer 1

thats incorrect

Answer 2

To determine how much the car will be worth in 7 years, we can analyze the given equation \( v(t) = 40,000(0.89)^t \).

In this context:
- \( v \) represents the value of the car after \( t \) years.
- \( t \) represents the number of years since the car was purchased, with \( t \) being in the range of \( 0 \) to \( 7 \) (inclusive, as we want to find the value specifically at year 7).

Now we can evaluate the appropriate ranges:
- **Option #1: \( 0 < t \leq 40,000 \)** — This option does not make sense for this context since \( t \) is a time in years and should not exceed 7.
- **Option #2: \( 0 < v \leq 40,000 \)** — This option is misleading because while the value \( v \) can never exceed the purchase price of $40,000, after a certain amount of time it will be less than the initial value; it does not describe the range accurately after depreciation.
- **Option #3: \( 0 < t \leq 7 \)** — This option correctly describes the range of time \( t \) over which we want to evaluate the car's depreciation.
- **Option #4: \( 0 < v \leq 7 \)** — This option is incorrect as it suggests \( v \) would be in this range and does not apply to the actual car's value.

Therefore, the most appropriate range that corresponds to Karen's inquiry about the car's value over the relevant time period is **Option #3: \( 0 < t \leq 7 \)**.