Problem 33 - Intermediate Algebra

Question

Use linear functions. The \(\mathrm{ABC}\) Car Rental uses the function \(f(x)=100\) for any daily use of a car up to and including 200 miles. For driving more than 200 miles per day, \(\mathrm{ABC}\) uses the function \(g(x)=100+0.25(x-200)\) to determine the charges. How much would \(\mathrm{ABC}\) charge for daily driving of 150 miles? of 230 miles? of 360 miles? of 430 miles?

Step-by-Step Solution

Answer

150 miles: 100, 230 miles: 107.5, 360 miles: 140, 430 miles: 157.5

Step 1: Understanding the Problem

We need to find the charge for daily driving different mileage amounts: 150 miles, 230 miles, 360 miles, and 430 miles, based on the given cost functions.

Step 1: Calculate Charge for 150 Miles

Since 150 miles is less than or equal to 200 miles, we use the function \(f(x) = 100\). The charge is simply \(100\).

Step 2: Calculate Charge for 230 Miles

Since 230 miles is greater than 200 miles, we use the function \(g(x) = 100 + 0.25(x-200)\). Substitute \(x = 230\) into the function:\[g(230) = 100 + 0.25(230 - 200) = 100 + 0.25(30) = 100 + 7.5 = 107.5\].The charge is \(107.5\).

Step 3: Calculate Charge for 360 Miles

Since 360 miles is greater than 200 miles, we use \(g(x)\). Substitute \(x = 360\) into the function:\[g(360) = 100 + 0.25(360 - 200) = 100 + 0.25(160) = 100 + 40 = 140\].The charge is \(140\).

Step 4: Calculate Charge for 430 Miles

Since 430 miles is greater than 200 miles, we use \(g(x)\). Substitute \(x = 430\) into the function:\[g(430) = 100 + 0.25(430 - 200) = 100 + 0.25(230) = 100 + 57.5 = 157.5\].The charge is \(157.5\).

Key Concepts

Piecewise Functions

Piecewise functions represent different expressions for different parts of their domain. They are akin to having separate rules for different situations. In our context,
ABC Car Rental's charging system uses two distinct rules based on the miles driven:

• For any mileage up to and including 200 miles, the cost is represented by the constant function: \( f(x) = 100 \).
• For mileage exceeding 200 miles, the cost calculation changes and is represented by: \( g(x) = 100 + 0.25(x-200) \). This formula accounts for an additional charge of $0.25 per mile over 200.

Understanding piecewise functions allows us to cater to various conditions within a single mathematical expression. It helps customize equations to match real-world scenarios by setting different rules for different conditions.

Function Application

The application of a function means determining the output by substituting a specific input value into the function's formula. For the car rental scenario, using piecewise functions,
we decide first which function applies for each case:

• If the mileage is 200 or below, use \( f(x) = 100 \) because the formula does not change with mileage.
• If the mileage is more than 200, use \( g(x) = 100 + 0.25(x-200) \), where you calculate an additional charge for each mile beyond 200.

For example, if you drive 230 miles, you use \( g(230) = 100 + 0.25(230 - 200) = 107.5 \), applying the laws of substitution. This logical method simplifies complex scenarios by dividing them into simpler components handled by suitable functions.

Cost Calculation in Mathematics

Cost calculation employs functions to establish a relationship between a service (like mileage) and its total cost. These calculations often occur in segmented formats like the piecewise functions used by ABC Car Rental. This allows precise pricing adjustments.
Key Steps in Cost Calculation:

• Define the base cost: In our example, the basic charge is \(100\) for up to 200 miles.
• Evaluate additional conditions: If usage exceeds a threshold, apply an extra cost tailored by the additional function \( g(x) \).
• Calculate total cost by substituting real values of mileage into the right function.

Thus, each increment over the base threshold affects the total charge, ensuring the pricing model remains fair and aligned with resource consumption. This strategy is widely applicable in real-world pricing models, retail, transport services, and utility billing, where usage varies.