Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

There is a proliferation of telephone-call billing schemes. According to one scheme, a call to anywhere in the United States is billed at 50 cents for the rst three minutes and \(9.8\) cents per minute after that. Express the cost of a call as a function of its duration.nates.

Step-by-Step Solution

Answer

The cost \(C(t)\) of a call of duration \(t\) is given by the piecewise function \(C(t) = \begin{cases} 0.50 & \text{if } t\leq3,\\ 0.50 + 0.098*(t-3)& \text{if } t>3. \end{cases}\)

Step 1: Define the Function Under Conditions

We need to express the cost in dollars of a telephone call, as a function of its duration, \(t\), measured in minutes. We know that a call is billed at two rates: 50 cents for the first three minutes, and 9.8 cents per minute after that.

Step 1: Expression for the cost of a 3 minute or lesser duration call

If the duration of the telephone call is 3 minutes or less, the cost would be fixed to $0.50, since that is the fixed price for calls that last up to three minutes. Mathematically, it would be represented as \(C(t)=0.50\) for \(t \leq 3\).

Step 2: Expression for the cost of a call exceeding 3 minutes in duration

If the duration of the telephone call is more than 3 minutes, the initial 3 minutes would be billed at $0.50, and every minute exceeding 3 minutes would be billed at $0.098. To calculate the final cost, we would multiply the number of minutes over 3 by $0.098 and add the initial $0.50. Mathematically it would be represented as \(C(t)=0.50 + 0.098*(t-3)\) for \(t > 3\).

Step 3: Final Piecewise Function

Combining these two conditions, we have a piecewise function representing the cost of a telephone call depending on its duration. This function is as follows: \[C(t) = \begin{cases} 0.50 & \text{if } t\leq3,\\ 0.50 + 0.098*(t-3)& \text{if } t>3. \end{cases}\]

Key Concepts

Mathematical Functions

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs. Each input is related to exactly one output. Functions are often represented as equations, such as the one described in a telephone-call billing scheme.

A telephone-call cost function is a practical example that shows how mathematics applies to real-world situations. In this specific scenario, the cost of the call is determined by its duration, making the duration (time in minutes) the input and the cost (in dollars) the output. This relationship is depicted with a function because each duration corresponds to one definitive cost, hence fulfilling the fundamental definition of a mathematical function.

Function Representation

To represent functions, mathematicians often use equations or graphical depictions. In the case of the telephone billing example, the function is expressed using a piecewise format. This is because the billing rates change after a certain point in time (3 minutes).

The piecewise function is divided into two scenarios: one for calls lasting 3 minutes or less and one for calls lasting longer than 3 minutes with different rates for each scenario. Piecewise functions are excellent tools for representing such situations because they clearly delineate where and how the function's formula changes.

Rate of Change

The concept of 'rate of change' in mathematics refers to how a quantity changes in relation to another. In the context of our telephone billing example, the rate of change appears when the call exceeds 3 minutes. For the first three minutes, there is no rate of change as the cost is a flat rate of 50 cents. However, after the third minute, the rate of change is 9.8 cents per minute. This means that for every additional minute spent on the phone, the cost increases by 9.8 cents.

Understanding the rate of change is crucial when dealing with piecewise functions because it helps you anticipate how the function's output (cost) will vary with an increase or decrease in the input (time). Rates of change can be constant, as in the flat rate for the first three minutes, or variable, as in the per-minute charge after three minutes.