A car depreciates according to the model V(t)≈19,000(0.719)t+700.
Based on this model, which statements are true?
There is more than one correct answer. Select all correct answers.
1.The approximate depreciation rate is about 28.1% per year.
2.The approximate depreciation rate is about 71.9% per year.
3.The original value of the car was $19,700.
4.The approximate value of the car will never drop below $700.
5.The approximate value of the car will never drop below $18,000.
6.The original value of the car was $700.
the answers to this are:
4: The approximate value of the car will never drop below $700
3: The original value of the car was $19,700
1: The approximate depreciation rate is about 28.1% per year
To determine which statements are true, let's analyze the model V(t) = 19,000(0.719)^t + 700:
1. The approximate depreciation rate is about 28.1% per year. [False]
The formula does not explicitly indicate the depreciation rate, so this statement cannot be determined from the given model.
2. The approximate depreciation rate is about 71.9% per year. [False]
The model does not offer a depreciation rate of 71.9% per year. This statement is not true.
3. The original value of the car was $19,700. [False]
The original value of the car can be determined by substituting t = 0 into the equation:
V(0) = 19,000(0.719)^0 + 700 = 19,000(1) + 700 = 19,000 + 700 = $19,700.
Therefore, the original value of the car was $19,700, and this statement is true.
4. The approximate value of the car will never drop below $700. [True]
The constant term in the equation is 700. As t increases, the exponential term (0.719)^t gradually decreases, but it will never reach zero. Thus, the car's value will never drop below $700, making this statement true.
5. The approximate value of the car will never drop below $18,000. [False]
From the equation V(t) = 19,000(0.719)^t + 700, as t increases, the exponential term (0.719)^t gradually diminishes. Eventually, it will approach zero, causing the total value V(t) to decrease and drop below $18,000. Therefore, this statement is false.
6. The original value of the car was $700. [False]
As previously calculated, the original value of the car was $19,700. This statement is not true.
Based on the given model, the statements that are true are:
- Statement 3: The original value of the car was $19,700.
- Statement 4: The approximate value of the car will never drop below $700.
To find the correct statements, let's analyze the given model V(t)≈19,000(0.719)t+700.
1. The approximate depreciation rate is about 28.1% per year.
To find the depreciation rate, we need to determine the factor by which the value changes each year. In this case, the factor is (0.719). To convert this factor into a percentage, we subtract it from 1 and multiply by 100.
Depreciation rate = (1 - 0.719) * 100 ≈ 28.1%
So, statement 1 is true.
2. The approximate depreciation rate is about 71.9% per year.
This statement is not true. The factors (0.719) corresponds to a depreciation rate of approximately 28.1% per year, not 71.9%. Therefore, statement 2 is false.
3. The original value of the car was $19,700.
To find the original value of the car, we need to substitute t = 0 into the model and solve for V(0).
V(0) ≈ 19,000(0.719)^0 + 700
V(0) ≈ 19,000 + 700
V(0) ≈ 19,700
So, statement 3 is true.
4. The approximate value of the car will never drop below $700.
According to the model, the constant term is $700. As t increases, the value may decrease due to the depreciation factor, but it will never drop below $700. Therefore, statement 4 is true.
5. The approximate value of the car will never drop below $18,000.
This statement is not true. As stated in the previous explanation, the constant term is $700, not $18,000. Therefore, statement 5 is false.
6. The original value of the car was $700.
Again, by substituting t = 0 into the model, we can find the original value:
V(0) ≈ 19,000(0.719)^0 + 700
V(0) ≈ 19,000 + 700
V(0) ≈ 19,700
Therefore, statement 6 is false. The original value of the car was $19,700, not $700.
So, the correct statements are:
1. The approximate depreciation rate is about 28.1% per year.
3. The original value of the car was $19,700.
4. The approximate value of the car will never drop below $700.