91Ó°ÊÓ

The concept of a cost function is central to understanding many business and financial calculations. A cost function models and describes how costs change with different levels of activity—in this case, the number of calling minutes over a cellphone plan. It gives us a way to calculate the total cost associated with a specific level of service usage.
In the exercise provided, the cost function is given by:

• \( C(t) = 20 + 0.40(t-60) \)

This cost function indicates that the customer pays a fixed monthly fee of \(20, along with \)0.40 for each additional minute above 60 minutes—and only above 60 minutes. The function is linear, meaning the rate of cost increase is constant based on usage. By understanding this function, we can predict and calculate the total charges for different scenarios without recalibration for each minute used over 60.

A piecewise function is made up of multiple sub-functions, each applying to a certain interval of the main function's domain. Although not explicitly mentioned, the cost function from the exercise can be considered as part of a larger piecewise function.Typically, the cost function given can be expressed in two parts:- Free for the first 60 minutes (a constant \(20 fee in practical terms)- \)0.40 per minute for each minute beyond 60This breakdown shows why the function is not relevant for \( t \leq 60 \); costs only vary when minutes surpass the threshold of 60. The piece "$0.40(t-60)" applies only when \( t > 60 \). Therefore, understanding this approach allows for proper evaluation when different conditions or intervals are considered. Cost functions behaving differently over different intervals reflect real-world scenarios, such as tiered pricing plans.

Evaluating a function means finding the value of the function at a given input or solving for specific conditions. In the provided exercise, you were tasked with evaluating the cost function at \( t = 100 \).

Evaluating a function involves plugging in the value into the function and solving it step by step. When \( t = 100 \), the function becomes:

• \( C(100) = 20 + 0.40 \, (100 - 60) \)

Simplifying gives:

• \( C(100) = 20 + 0.40 \, (40) \)
• \( C(100) = 20 + 16 \)
• \( C(100) = 36 \)

This calculation means that for 100 calling minutes, the monthly cost totals $36. This function evaluation process helps us project costs effectively and adjust service usage according to budget constraints.