Problem 82 - Algebra and Trigonometry

Question

A taxi company charges \(\$ 2.00\) for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost \(C\) (in dollars) of a ride as a piecewise-defined function of the distance \(X\) traveled (in miles) for \(0

Step-by-Step Solution

Answer

The cost function is \(C(X) = 2.00\) for \(0 < X \leq 1\) and \(C(X) = 2X\) for \(1 < X < 2\).

Step 1: Understanding the components of the charge

The taxi company charges \(\\(2.00\) for the first mile (or part of a mile). Afterward, every succeeding tenth of a mile (or part) is charged at \(\\)0.20\). We are to express these charges in a piecewise function based on the distance traveled, \(X\), where \(0 < X < 2\).

Step 2: Express cost for X less than or equal to 1 mile

For the first mile, the function is constant at \\(2.00 since the cost includes the first mile charge: \(C(X) = 2.00\) for \(0 < X \leq 1\). This means any travel distance within the first mile only incurs a charge of \\)2.00.

Step 3: Calculate cost for 1 < X < 2 miles

For numbers larger than 1, each additional tenth of a mile costs \(\\(0.20\). If \(X\) denotes the total distance traveled, then \(X - 1\) is the distance charged at \(\\)0.20\) per tenth mile. There are \(10(X-1)\) parts of a tenth mile in \(X-1\). Calculate cost as \(C(X) = 2.00 + 0.20 \times 10(X-1)\). Simplifying the function gives \(C(X) = 2.00 + 2(X-1) = 2X\) for \(1 < X < 2\).

Step 4: Combine into a piecewise function

The complete piecewise function becomes:\[C(X) = \begin{cases} 2.00, & 0 < X \leq 1 \2X, & 1 < X < 2\end{cases}\]This defines the cost structure for different segments of travel distance within \(0 < X < 2\) miles.

Step 5: Sketch the Graph

The graph of the piecewise function consists of two parts. For \(0 < X \leq 1\), plot a horizontal line at \(Y = 2.0\), representing the fixed cost. For \(1 < X < 2\), plot a line with slope 2 starting from the point (1, 2). The second segment extends linearly up to (2, 4). Merge both segments smoothly. This depicts the transition from a flat rate to a variable rate.

Key Concepts

Function of Distance

A function of distance in this context refers to how the cost of a taxi ride changes depending on the distance traveled. For the range between 0 and 2 miles, the function calculates the total cost by evaluating how far you've traveled from the starting point. In the problem, the distance affects the cost in two main segments. Each part of the distance has a different charge rate, first as a flat rate and secondly as a variable rate. This means that the cost is directly tied to how much distance you've covered in your journey, making it a distance-based function.

Taxi Fare Calculation

Taxi fare calculation is based on a specific pricing structure that varies with distance. Initially, you pay a flat fare for the first mile. The problem specifies a fixed rate of \(\$2.00\) for the first mile or part of it, regardless of the exact number of yards or quarters of the mile.As you exceed the first mile, the pricing structure shifts to a varying charge, where an additional \(0.20\) is charged for each subsequent tenth of a mile. Consequently, this method of calculating fare ensures the cost aligns with the actual distance traveled, offering fairness and precision in fare determination.

Cost Function

A cost function helps to predict the total price associated with a service, based on specific input parameters. In the taxi fare problem, the cost function is defined piecewise: - For travel within the first mile, the cost remains constant at \(\$2.00\).- Beyond 1 mile, up to the next, the cost increments at \(0.20\) per tenth of a mile, yielding the formula \(C(X) = 2X\) for \(1 < X < 2\).This function is an essential tool for determining taxi expenses, giving us a step-by-step calculation of costs linked with travel distances.

Piecewise Functions in Algebra

Piecewise functions are a fundamental concept in algebra known for handling situations where a function's rule changes based on the input value. They allow for different expressions over separate intervals of a domain, which is the case with the cab fare exercise.The piecewise function here is expressed as:\[C(X) = \begin{cases} 2.00, & 0 < X \leq 1 \ 2X, & 1 < X < 2 \end{cases}\]This clearly breaks down the fare into distinct messages depending on distance traveled, with each piece applying under particular conditions of X, the total distance traveled.

Graphing Piecewise Functions

Graphing piecewise functions provides a visual representation of how these functions behave over different intervals. For the taxi fare problem, graphing this piecewise function involves plotting two distinct segments:- A flat horizontal line at \(Y = 2.0\) for the interval \(0 < X \leq 1\), indicating the fixed cost for the initial mile.- A sloped line, starting from the point (1, 2), increasing with a slope of 2, extending to (2, 4) within the interval \(1 < X < 2\).This graph effectively illustrates the transition from a constant cost to a linearly increasing one, making it easier for students to conceptualize how piecewise functions react to input changes.