you decide to buy a car for 25000. each year your car depreciates in value by 5% from its original value. After 15 years though, your used car will only be worth $500 to a dealership.
Write a piece wise function for C(t), car value over time, for this scenario
Graph the piecewise function
If you pay $6000 a year for your car, when will it be a good to trade in your car?
V(t) = Po - Po*r*t
V(t) = 25,000 - 25000*0.05*t
V(t) = 25000(1-0.05t)
To create a piecewise function for the car value over time, we need to consider the different time intervals and the corresponding car values.
Let's break it down into three intervals:
1. From year 0 to year 15: The car depreciates by 5% each year.
2. After year 15: The car is worth a constant value of $500.
3. After you start paying $6000 a year for a new car: We need to find when it becomes more economical to trade in the car.
Let's define the piecewise function:
For 0 ≤ t ≤ 15, C(t) = $25,000 * (1 - 0.05)^t
From year 16 onwards, t > 15, C(t) = $500
To find when it becomes more economical to trade in the car, we need to compare the cost of paying $6000 a year with the cost of keeping the car. So, we set up an equation:
$6000 * t = $25,000 * (1 - 0.05)^15
Simplifying, we have:
$6000 * t = $500
Dividing both sides by $6000, we get:
t = $500 / $6000
Now we can calculate the value of t:
t ≈ 0.0833 years
To convert this into months, we multiply by 12:
t ≈ 0.0833 * 12
t ≈ 0.9996 months
Therefore, it will be a good time to trade in your car after approximately 1 month.
To graph the piecewise function, we plot the car value over time. The y-axis represents the car value in dollars, and the x-axis represents time in years. The first part of the graph is a decreasing curve, and after year 15, it becomes a horizontal line at $500.