Application of derivative in road taxation worked examples
Example 1:
Suppose a car is traveling along a road at a constant speed of 60 miles per hour. The road tax for this particular car is based on its speed, with a rate of $1 per mile per hour.
Let x represent the distance traveled in miles, and y represent the road tax in dollars.
Since the car is traveling at a constant speed of 60 miles per hour, the distance x is increasing at a constant rate of 60 miles per hour. Therefore, we have dx/dt = 60.
We want to find dy/dt, the rate at which the road tax is changing with respect to time. Using the chain rule, we have:
dy/dt = dy/dx * dx/dt
Since x represents the distance traveled and y represents the road tax, dy/dx represents the rate at which the road tax changes with respect to distance traveled.
dy/dx = 1 mile per hour
Now, plugging in the known values, we have:
dy/dt = (1 mile per hour) * (60 miles per hour) = 60 dollars per hour
Therefore, the road tax for this car is increasing at a rate of 60 dollars per hour.
Example 2:
Suppose a truck is traveling along a road, and its speed is given by the function v(t) = 20t + 10, where t represents time in hours and v(t) represents the speed in miles per hour. The road tax for this truck is based on its speed, with a rate of $2 per mile per hour.
Let x represent the distance traveled in miles, and y represent the road tax in dollars.
To find dy/dt, the rate at which the road tax is changing with respect to time, we need to find dx/dt, the rate at which the distance traveled is changing with respect to time.
Since the speed of the truck is given by v(t) = 20t + 10, we have dx/dt = v(t) = 20t + 10.
We want to find dy/dt, so we need to find dy/dx, the rate at which the road tax changes with respect to distance traveled.
dy/dx = 2 dollars per mile per hour
Using the chain rule, we have:
dy/dt = dy/dx * dx/dt
Plugging in the known values, we have:
dy/dt = (2 dollars per mile per hour) * (20t + 10 miles per hour)
Therefore, dy/dt = 40t + 20 dollars per hour.
This means that the road tax for the truck is increasing at a rate of 40t + 20 dollars per hour, where t is the time elapsed since the truck started traveling.