Period: Date: 2/25Charges: $5 For The First Hour, $10 For Up To Two Hours. Calculate The Dollar Cost Of Parking For $t$ Hours.$\[ \begin{tabular}{|c|c|} \hline $t$ (hours) & $G$ (cost In Dollars) \\ \hline 0 & 0
Introduction
Parking fees can be a significant expense for individuals who need to park their vehicles for extended periods. In this article, we will explore a mathematical approach to calculating the cost of parking based on the duration of parking. We will use a specific parking fee structure as an example to demonstrate how to calculate the cost of parking for a given number of hours.
Parking Fee Structure
The parking fee structure is as follows:
- $5 for the first hour
- $10 for up to two hours
This means that if you park for more than two hours, the cost will be $5 for the first hour and $10 for each additional hour.
Calculating the Cost of Parking
To calculate the cost of parking for a given number of hours, we need to use the following formula:
G(t) = 5 + 10(max(0, t - 1))
where G(t) is the cost of parking for t hours, and max(a, b) is the maximum of a and b.
Explanation of the Formula
The formula works as follows:
- If t is less than or equal to 1, the cost is simply $5, since you are only paying for the first hour.
- If t is greater than 1, the cost is $5 for the first hour and $10 for each additional hour. The max function is used to ensure that the cost is not negative.
- The formula is then simplified to G(t) = 5 + 10(t - 1), which is the final formula for calculating the cost of parking.
Example Calculations
Let's use the formula to calculate the cost of parking for different numbers of hours.
Example 1: Parking for 1 hour
- t = 1
- G(t) = 5 + 10(max(0, 1 - 1)) = 5 + 10(0) = 5
The cost of parking for 1 hour is $5.
Example 2: Parking for 2 hours
- t = 2
- G(t) = 5 + 10(max(0, 2 - 1)) = 5 + 10(1) = 15
The cost of parking for 2 hours is $15.
Example 3: Parking for 3 hours
- t = 3
- G(t) = 5 + 10(max(0, 3 - 1)) = 5 + 10(2) = 25
The cost of parking for 3 hours is $25.
Conclusion
In this article, we have explored a mathematical approach to calculating the cost of parking based on the duration of parking. We have used a specific parking fee structure as an example to demonstrate how to calculate the cost of parking for a given number of hours. The formula G(t) = 5 + 10(max(0, t - 1)) can be used to calculate the cost of parking for any number of hours.
References
- [1] Parking Fee Structure. (n.d.). Retrieved from https://www.example.com/parking-fee-structure
Discussion
This problem can be solved using a variety of mathematical techniques, including algebra and calculus. The formula G(t) = 5 + 10(max(0, t - 1)) is a simple and effective way to calculate the cost of parking for a given number of hours.
Mathematical Derivation
The formula G(t) = 5 + 10(max(0, t - 1)) can be derived using the following steps:
- Define the cost of parking for the first hour as $5.
- Define the cost of parking for each additional hour as $10.
- Use the max function to ensure that the cost is not negative.
- Simplify the formula to G(t) = 5 + 10(t - 1).
Code Implementation
The formula G(t) = 5 + 10(max(0, t - 1)) can be implemented in a variety of programming languages, including Python and MATLAB. Here is an example implementation in Python:
def calculate_parking_cost(t):
return 5 + 10(max(0, t - 1))
This function takes a single argument t, which represents the number of hours, and returns the cost of parking for that number of hours.
Conclusion
Q: What is the cost of parking for 0 hours?
A: The cost of parking for 0 hours is $0.
Q: What is the cost of parking for 1 hour?
A: The cost of parking for 1 hour is $5.
Q: What is the cost of parking for 2 hours?
A: The cost of parking for 2 hours is $15.
Q: What is the cost of parking for 3 hours?
A: The cost of parking for 3 hours is $25.
Q: How do I calculate the cost of parking for a given number of hours?
A: To calculate the cost of parking for a given number of hours, you can use the formula G(t) = 5 + 10(max(0, t - 1)), where t is the number of hours.
Q: What is the max function in the formula?
A: The max function is used to ensure that the cost is not negative. If t is less than or equal to 1, the max function returns 0, and the cost is simply $5. If t is greater than 1, the max function returns t - 1, and the cost is $5 for the first hour and $10 for each additional hour.
Q: Can I use this formula to calculate the cost of parking for any number of hours?
A: Yes, you can use this formula to calculate the cost of parking for any number of hours.
Q: Is there a simpler way to calculate the cost of parking?
A: Yes, you can simplify the formula to G(t) = 5 + 10(t - 1), which is a more straightforward way to calculate the cost of parking.
Q: Can I implement this formula in a programming language?
A: Yes, you can implement this formula in a variety of programming languages, including Python and MATLAB.
Q: What is the code implementation of this formula?
A: Here is an example implementation in Python:
def calculate_parking_cost(t):
return 5 + 10(max(0, t - 1))
This function takes a single argument t, which represents the number of hours, and returns the cost of parking for that number of hours.
Q: Can I use this formula to calculate the cost of parking for a specific parking fee structure?
A: Yes, you can use this formula to calculate the cost of parking for a specific parking fee structure. Simply replace the values in the formula with the corresponding values for your parking fee structure.
Q: Is there a mathematical derivation of this formula?
A: Yes, there is a mathematical derivation of this formula. The formula can be derived using the following steps:
- Define the cost of parking for the first hour as $5.
- Define the cost of parking for each additional hour as $10.
- Use the max function to ensure that the cost is not negative.
- Simplify the formula to G(t) = 5 + 10(t - 1).
Q: Can I use this formula to calculate the cost of parking for a variety of parking fee structures?
A: Yes, you can use this formula to calculate the cost of parking for a variety of parking fee structures. Simply replace the values in the formula with the corresponding values for your parking fee structure.
Conclusion
In this article, we have answered frequently asked questions about calculating the cost of parking using the formula G(t) = 5 + 10(max(0, t - 1)). We have also provided a code implementation of this formula in Python and a mathematical derivation of the formula.