Application of Derivative in road taxation equation and worked example
An application of derivatives in road taxation can be found in calculating the tax owed by a vehicle based on its weight. The tax rate is typically determined by the weight of the vehicle, where heavier vehicles pay higher taxes.
In this scenario, let's assume that the tax rate is determined by the weight of a vehicle, W (in kilograms). The tax rate function, R(W), is given as:
R(W) = 0.05W + 100
To calculate the tax owed by a specific vehicle, we need to find the derivative of the tax rate function with respect to the weight, dR/dW. This derivative represents the rate of change of the tax rate with respect to the weight.
dR/dW = 0.05
Once we have the derivative, we can determine how the tax rate changes as the weight of the vehicle increases. In this case, the tax rate increases by 0.05 units for every unit increase in weight.
Now, let's work through an example:
Suppose we have a vehicle that weighs 1500 kilograms. We want to calculate the tax owed by this vehicle based on the given tax rate function.
First, we calculate the derivative of the tax rate function:
dR/dW = 0.05
Since the derivative is a constant, it means that the tax rate is not affected by the weight of the vehicle.
Next, we substitute the weight of the vehicle into the tax rate function:
R(1500) = 0.05(1500) + 100
= 75 + 100
= 175
Therefore, the tax owed by a vehicle weighing 1500 kilograms is 175 units.
This is just one example of how derivatives can be applied in road taxation. By understanding the rate of change of the tax rate with respect to different factors, such as weight, governments can calculate fair taxation policies.