Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

The cost of parking in a metered lot is \(\$ 1.00\) for the first hour and \(\$ 0.50\) for each additional hour or portion of an hour. (a) \(\mathrm{A}\) customer needs a model for the cost \(C\) of parking in the metered lot for \(t\) hours. Which of the following is the appropriate model? \(C_{1}(t)=1+0.50[t-1]\) \(C_{2}(t)=1-0.50[-(t-1)]\) (b) Use a graphing utility to graph the appropriate model. Estimate the cost of parking in the metered lot for 7 hours and 10 minutes.

Step-by-Step Solution

Answer

The appropriate model is \(C_{1}(t)=1+0.50(t-1)\). The estimated cost of parking for 7 hours and 10 minutes based on this model is approximately $4.08.

Step 1: Choose the Correct Model

The first hour of parking costs $1, and then the cost increases by $0.50 for each additional hour or a fraction of an hour. Therefore, the model reflects the initial cost of $1 plus $0.50 times the number of hours minus 1 (because of the first hour). Thus the first model, \(C_{1}(t)=1+0.50(t-1)\), is the correct mathematical description of the cost of parking, where \(t\) is the time of parking in hours.

Step 2: Apply the Model for Estimation

The second part of the problem asks to estimate the cost for 7 hours and 10 minutes using the chosen model. Now, convert 10 minutes into hours (10/60 = 0.167 hours), then add this to the 7 hours, for a total of 7.167 hours. Substitute \(t = 7.167\) into the model \(C_{1}(t)\), and we get \(C_{1}(7.167) = 1 + 0.50(7.167 - 1)\). Calculate this expression gives the cost of parking for 7 hours and 10 minutes.

Key Concepts

Linear Piecewise Functions

Understanding linear piecewise functions is essential for students tackling algebra problems that involve separate equations for different intervals.

A piecewise function is a function composed of multiple sub-functions, each defined on a certain interval. The function takes a different form based on the input value. For example, consider a function that charges a different rate based on the hours worked or a taxi fare that changes with distance covered.

Application of Linear Piecewise Functions

In the parking lot cost example, the first hour costs a flat rate, and the subsequent hours have a different rate, which implies a different equation after the first hour. Thus, the correct model is a linear piecewise function, adjusted for these intervals. To visualize them, plot the function piece by piece on the coordinate plane, using separate linear equations for each segment of the graph.

Graphing Utilities

Graphing utilities, like graphing calculators or computer software, are indispensable tools for visualizing mathematical concepts, particularly linear piecewise functions.

These utilities allow students to input the different equations for variable intervals and view the corresponding piecewise graph.

Using Graphing Utilities Effectively

When graphing the model for parking costs, students should check the settings to ensure they're configured to display piecewise functions accurately. Graphing the selected model helps to estimate the parking cost over continuous periods, reinforcing the understanding of how the function behaves over time.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world scenarios. It serves as a bridge between mathematics and practical applications, allowing for predictions and decisions based on mathematical calculations.

For instance, the cost of parking in a metered lot can be modeled using a linear piecewise function to reflect different pricing for the first hour and subsequent hours.

Key Aspects of Mathematical Modeling

To create an effective model, identify all variables and constants, like the fixed initial charge and the variable hourly rate. Translate these elements into algebraic expressions to forecast outcomes, such as estimating the cost of parking for any given time.

Time Conversion in Algebra

Whenever algebraic problems involve time, students may need to convert units for consistency and clarity, which is referred to as time conversion in algebra.

Time can be measured in hours, minutes, and seconds, but for algebraic functions, it's often most convenient to express all time in a single unit, typically hours.

Steps for Time Conversion

To solve the parking cost problem, convert 10 minutes into hours by dividing by 60, since there are 60 minutes in an hour. This standard conversion allows for accurate substitution into algebraic models and is a crucial step in solving time-related algebra problems.