After t years, the value of a car that originally cost $16,000 depreciates so that each year is is worth 3/4 of it's value for the precious year. Find a model for V(t), the value of the car after t years. Sketch a graph of the model and determine the value of the car 4 years after it was purchased.
2399000
4 years
To find a model for V(t), we need to determine the relationship between the value of the car and the number of years that have passed.
Given that the value of the car after each year is 3/4 of its value from the previous year, we can express the relationship as follows:
V(t) = (3/4)^t * 16,000
This model takes into account the initial value of $16,000 and the fact that the value decreases by 3/4 each year.
To sketch a graph of the model, we can plot the value of the car (V) on the y-axis and the number of years (t) on the x-axis. Since the value decreases over time, the graph will be a decreasing exponential function.
Now, let's determine the value of the car 4 years after it was purchased:
V(4) = (3/4)^4 * 16,000
= (81/256) * 16,000
≈ 5,000
So, the value of the car 4 years after it was purchased is approximately $5,000.
To find a model for V(t), the value of the car after t years, we need to determine the general equation that represents the depreciation of the car.
The given information tells us that each year, the car is worth 3/4 of its value from the previous year. This means that the value of the car after one year can be expressed as 3/4 times its value at the previous year.
Let V(t) represent the value of the car after t years. We can write the equation as follows:
V(t) = (3/4) * V(t-1)
Now, to form a model for V(t), we need an initial condition. The original cost of the car is given as $16,000, so our initial condition is:
V(0) = $16,000
Using this, we can recursively calculate the value of the car after each year.
To determine the value of the car after 4 years, plug t = 4 into the model equation:
V(4) = (3/4) * V(3)
= (3/4) * (3/4) * V(2)
= (3/4) * (3/4) * (3/4) * V(1)
= (3/4)^4 * V(0)
= (3/4)^4 * $16,000
Now let's calculate this:
V(4) = (3/4)^4 * $16,000
≈ 0.3164 * $16,000
≈ $5,062.50
Therefore, the value of the car 4 years after it was purchased is approximately $5,062.50.
Now let's sketch a graph of the model. We will plot the value of the car (V) on the y-axis and the number of years (t) on the x-axis. The curve will show the exponential decay of the value of the car over time.
Note: Without specific intervals and units for the axes, the scale of the graph cannot be determined accurately.