Problem 132 - Precalculus Essentials

Question

132\. The function \(C(t)=20+0.40(t-60)\) describes the monthly cost, \(C(t),\) in dollars, for a cellphone plan for the calling minutes, where \(t>60\). Find and interpret \(C(100)\).

Step-by-Step Solution

Answer

The cost, \(C(100)\), ends up being $36. This means that for 100 calling minutes, the cellphone plan would cost $36 per month.

Step 1: Substitute the value of t in the function

First, put \(t=100\) into the cost function. We get: \(C(100)=20+0.40(100-60)\)

Step 2: Simplify the function

Next, simplify the function that results from the substitution. That would look like this: \(C(100)=20+0.40(40)\)

Step 3: Evaluate the function

Finally, calculate the cost by multiplying 0.40 by 40 and then by adding 20. This results in: \(C(100)=20+\(16)=36\)

Key Concepts

Function Evaluation

Understanding how to evaluate a function is essential in precalculus, mainly because it sets the foundation for dealing with more complex mathematical concepts. Function evaluation is the process of determining the output of a function for a particular input.

Let's take the given function from the exercise, \(C(t)=20+0.40(t-60)\). To evaluate this function for \(t=100\), you substitute the value of 100 for \(t\). In practical terms, this represents finding the monthly cost for using 100 minutes on a certain cellphone plan. After substitution, simplification, and performing the arithmetic operations as shown in the steps, we find that \(C(100)=36\) dollars, meaning the cost is 36 dollars when a person uses 100 minutes of calling time.

Piecewise Functions

A piecewise function is a type of function that is defined by multiple sub-functions, each with its own domain. Each piece is applicable to different intervals or conditions in the domain of the function.

In the context of the given exercise, even though the function \(C(t)\) for the cellphone plan isn't explicitly described as piecewise, it implicitly behaves like one. This is due to the condition \(t>60\). It implies that the cost function changes its structure for \(t\) values less than or equal to 60 and therefore, would have a different equation or condition for those values. A full piecewise representation would provide cost formulae for all ranges of \(t\), not just when \(t>60\).

Interpreting Function Results

Interpreting function results involves not just calculating an answer but understanding what the results mean in a real-world context.

For the function \(C(100)=36\) from the exercise, the interpretation is as follows: If a person uses 100 minutes on their cellphone plan, the total cost for the month is \(36. This cost is based on a flat rate of \)20, with an additional variable cost that depends on the number of minutes used beyond the base 60 minutes. Understanding these results allows one to grasp the implications of the function in practical scenarios, such as budgeting for cellphone expenses.

Linear Cost Function

In precalculus, linear cost functions are used to model situations where the total cost is a linear function of some variable quantity. They are represented as \(C(x)=mx+b\), where \(m\) is the variable cost per unit, \(b\) is the fixed cost, and \(x\) represents the quantity.

In our exercise, the cost function \(C(t)=20+0.40(t-60)\) models the cost of a cellphone plan. The fixed cost here is \(20, and the variable cost is \)0.40 per minute above 60 minutes. This type of function is crucial for understanding how costs scale with usage and can be applied to various economic and business contexts, such as predicting expenses, setting budget plans, or analyzing consumer behavior.