Question
The value of a car decreases over time according to the equation V = 25,000(0.80)ⁿ, where V represents the value of the car in dollars, and n represents the number of years since the car was purchased.
Complete the statements.
In the equation, the number 25,000 represents the
of the car. The value of the car is
at a rate of
per year.
What would V = 30,000 mean in the context of this problem?
Is it possible for V = 30,000 in context of this problem?
What does t = 3 mean in context of this problem?
Complete the statements.
In the equation, the number 25,000 represents the
of the car. The value of the car is
at a rate of
per year.
What would V = 30,000 mean in the context of this problem?
Is it possible for V = 30,000 in context of this problem?
What does t = 3 mean in context of this problem?
Answers
GPT-4o mini
Let's analyze the equation \( V = 25,000(0.80)^n \) step by step.
1. In the equation, the number **25,000 represents the initial value of the car** (i.e., the value of the car when it was first purchased).
2. The value of the car is **decreasing** at a rate of **20% per year** because the car retains 80% of its value each year (expressed as the factor \( 0.80 \)).
3. For the statement regarding \( V = 30,000 \):
- In the context of this problem, \( V = 30,000 \) would imply that the value of the car is $30,000 at some point in time. However, since the equation specifies \( V \) as decreasing from an initial value of $25,000, it indicates that \( V = 30,000 \) is **not possible** within the context of this problem, as the value can never exceed the initial value after any number of years.
4. For the statement regarding \( n = 3 \):
- The value \( n = 3 \) means that **three years have passed since the car was purchased**. This corresponds to the calculation of the car's value at that time using the equation \( V = 25,000(0.80)^3 \).
Overall:
- The initial value of the car is 25,000.
- The car's value decreases at a rate of 20% per year.
- \( V = 30,000 \) is not possible because the car's value can only decrease from the original 25,000.
- \( t = 3 \) means three years after the purchase of the car.
1. In the equation, the number **25,000 represents the initial value of the car** (i.e., the value of the car when it was first purchased).
2. The value of the car is **decreasing** at a rate of **20% per year** because the car retains 80% of its value each year (expressed as the factor \( 0.80 \)).
3. For the statement regarding \( V = 30,000 \):
- In the context of this problem, \( V = 30,000 \) would imply that the value of the car is $30,000 at some point in time. However, since the equation specifies \( V \) as decreasing from an initial value of $25,000, it indicates that \( V = 30,000 \) is **not possible** within the context of this problem, as the value can never exceed the initial value after any number of years.
4. For the statement regarding \( n = 3 \):
- The value \( n = 3 \) means that **three years have passed since the car was purchased**. This corresponds to the calculation of the car's value at that time using the equation \( V = 25,000(0.80)^3 \).
Overall:
- The initial value of the car is 25,000.
- The car's value decreases at a rate of 20% per year.
- \( V = 30,000 \) is not possible because the car's value can only decrease from the original 25,000.
- \( t = 3 \) means three years after the purchase of the car.