Calculate Shipping Cost With F(x) = 5[x/10] + 5

by Alex Johnson 48 views

Let's break down how to calculate shipping costs using the function provided: f(x) = 5[x/10] + 5. This function might look a little intimidating at first, but we'll go through it step by step. The key here is understanding the square brackets, which represent the floor function. The floor function, denoted by [x], gives you the greatest integer less than or equal to x. For instance, [3.2] = 3, [7.9] = 7, and [5] = 5. Now, we'll apply this function to a series of distances to determine the corresponding shipping charges. This involves substituting the given distances into the function and performing the calculations. Each distance will yield a unique shipping cost based on the formula's parameters. Let's dive in and calculate the charges for each specified distance to understand how this function works in practice. By the end of this, you'll be a pro at using floor functions to calculate shipping costs!

Understanding the Function

Before we jump into calculations, let's make sure we fully grasp what the function f(x) = 5[x/10] + 5 is telling us. Here, 'x' represents the distance in miles, and f(x) will give us the shipping charge in some currency (let's assume it's dollars for simplicity). The core of the function lies in the term [x/10]. This is where the floor function comes into play. We're dividing the distance 'x' by 10, and then taking the floor of the result. This essentially groups distances into segments of 10 miles. For every 10 miles (or portion thereof), the base cost will increase. After finding the floored value, we multiply it by 5. This suggests that for every 10-mile segment, there's a $5 increment in the shipping cost. Finally, we add 5 to the result. This likely represents a fixed base charge, regardless of the distance. So, no matter how short the distance, there will always be a minimum charge of $5. To visualize this, imagine shipping something within a mile – you'd still pay this base charge. Understanding these components – the distance segments, the incremental cost, and the base charge – is crucial for accurately calculating shipping costs for various distances. Let's move on to applying this knowledge to specific distances.

Calculating Shipping Charges for Different Distances

Now, let's get to the heart of the matter and calculate the shipping charges for the provided distances using our function, f(x) = 5[x/10] + 5. We'll tackle each distance one by one, showing the step-by-step process so you can see exactly how the function works in practice.

1. 83.41.2 Miles

  • First, we substitute x = 41.2 into the function: f(41.2) = 5[41.2/10] + 5
  • Next, we divide 41.2 by 10, which gives us 4.12.
  • Now, we apply the floor function: [4.12] = 4
  • Substitute this value back into the equation: 5 * 4 + 5
  • Finally, we calculate the result: 20 + 5 = 25
  • So, the shipping charge for 41.2 miles is $25.

2. 30 Miles

  • Substitute x = 30 into the function: f(30) = 5[30/10] + 5
  • Divide 30 by 10: 30/10 = 3
  • Apply the floor function: [3] = 3
  • Substitute back into the equation: 5 * 3 + 5
  • Calculate the result: 15 + 5 = 20
  • Therefore, the shipping charge for 30 miles is $20.

3. 14.5 Miles

  • Substitute x = 14.5: f(14.5) = 5[14.5/10] + 5
  • Divide 14.5 by 10: 14.5/10 = 1.45
  • Apply the floor function: [1.45] = 1
  • Substitute: 5 * 1 + 5
  • Calculate: 5 + 5 = 10
  • The shipping charge for 14.5 miles is $10.

4. 79.8 Miles

  • Substitute x = 79.8: f(79.8) = 5[79.8/10] + 5
  • Divide 79.8 by 10: 79.8/10 = 7.98
  • Apply the floor function: [7.98] = 7
  • Substitute: 5 * 7 + 5
  • Calculate: 35 + 5 = 40
  • The shipping charge for 79.8 miles is $40.

5. 100 Miles

  • Substitute x = 100: f(100) = 5[100/10] + 5
  • Divide 100 by 10: 100/10 = 10
  • Apply the floor function: [10] = 10
  • Substitute: 5 * 10 + 5
  • Calculate: 50 + 5 = 55
  • The shipping charge for 100 miles is $55.

6. 202.5 Miles

  • Substitute x = 202.5: f(202.5) = 5[202.5/10] + 5
  • Divide 202.5 by 10: 202.5/10 = 20.25
  • Apply the floor function: [20.25] = 20
  • Substitute: 5 * 20 + 5
  • Calculate: 100 + 5 = 105
  • The shipping charge for 202.5 miles is $105.

By working through these examples, you can see how the function effectively calculates shipping costs based on distance. The floor function ensures that costs are calculated in 10-mile increments, while the base charge provides a minimum fee for any shipment. Now you have a solid understanding of how to use this function!

Summarizing the Shipping Charges

Let's quickly recap the shipping charges we calculated for each distance. This will give us a clear overview of how the function f(x) = 5[x/10] + 5 works in practice.

  • For 41.2 miles, the shipping charge is $25.
  • For 30 miles, the shipping charge is $20.
  • For 14.5 miles, the shipping charge is $10.
  • For 79.8 miles, the shipping charge is $40.
  • For 100 miles, the shipping charge is $55.
  • For 202.5 miles, the shipping charge is $105.

As you can see, the shipping cost increases in steps, reflecting the 10-mile segments dictated by the floor function. This method provides a structured and predictable way to calculate shipping fees based on distance. The base charge of $5 ensures a minimum cost, while the incremental charge of $5 for each 10-mile segment (or part thereof) accounts for the distance traveled. This makes the function a practical tool for any shipping company looking to implement a fair and transparent pricing system. Now, let’s think about how this kind of function might be useful in other real-world scenarios.

Real-World Applications and Implications

The function f(x) = 5[x/10] + 5 is a simplified model, but it illustrates a common approach to calculating costs based on tiered systems. Beyond shipping, this type of function, incorporating the floor function, has several real-world applications. For example, consider parking fees: a parking garage might charge a flat fee for the first hour and then an additional fee for each subsequent hour or portion thereof. This can be modeled using a similar function, where the floor function helps to round up to the nearest hour. Another application is in taxation, where tax brackets often work in a tiered system. Income within a certain range is taxed at one rate, and income above that range is taxed at a higher rate. The floor function (or similar rounding mechanisms) can be used to determine which tax bracket applies to a given income. In telecommunications, data plans often have a base price for a certain amount of data, with additional charges kicking in for each gigabyte (or portion thereof) used beyond the limit. These tiered pricing structures are ubiquitous because they provide a balance between simplicity for the customer and cost recovery for the service provider. They allow for predictable pricing while accounting for varying levels of usage. Understanding the mathematical principles behind these systems, like the use of the floor function, helps consumers and businesses alike to make informed decisions. If you're interested in learning more about how mathematical functions are used in business and economics, you might find valuable resources on websites like Investopedia's Economics Basics.