Question

A cell phone company charges \(\$ 0.12\) for connecting a call plus \(\$ 0.08\) per minute or any part thereof (e.g., a phone call lasting 2 minutes and 5 seconds costs \(\$ 0.12+3 \times \$ 0.08\) ). Sketch a graph of the cost of making a call as a function of the length of time \(t\) that the call lasts. Discuss the continuity of this function.

Short answer

The cost function is discontinuous due to discrete jumps at each whole minute.

Step-by-step solution

Understanding the Cost Formula

First, recognize the structure of the costs involved. The total cost of a call, which we denote as \( C(t) \), is composed of a fixed connection fee of \( \\(0.12 \) plus \( \\)0.08 \) per minute or part thereof. Mathematically, this can be represented as \( C(t) = 0.12 + 0.08 \, \lceil t \rceil \), where \( \lceil t \rceil \) is the ceiling function that rounds up the time \( t \) to the next whole number of minutes.

Sketching the Graph

To sketch the graph, plot the cost \( C(t) \) as a function of time \( t \) in minutes. The graph consists of steps due to the ceiling function. For intervals between whole n-minute marks (like from \( t=1 \) minute to \( t<2 \) minutes), the cost remains constant. At each whole minute mark, the cost jumps up by \( \$0.08 \). Start at \( 0.12 \) at time \( t=0 \), jump to \( 0.20 \) at time \( t=1 \), and so forth.

Analyzing Continuity

Consider the nature of the graph plotted. The function \( C(t) = 0.12 + 0.08 \, \lceil t \rceil \) is discontinuous at every positive integer \( t \). This is because the cost jumps up by \( \$0.08 \) abruptly at each integer point without any intermediary values. Continuous functions have no jumps, hence this function is discontinuous wherever a jump occurs.

Key concepts

Cost Function

A cost function is a mathematical way to express the total cost associated with a certain process or action. In the context of our exercise, the cost function helps determine the total cost of making a phone call, which includes both fixed and variable components.
The cost function, denoted as \( C(t) \), presents the cost of a call lasting \( t \) minutes. Here, a fixed connection fee of \( \\(0.12 \) is charged initially for setting up the call. Additionally, there is a variable charge of \( \\)0.08 \) per minute that the call is in progress.
In our specific piecewise function form, this variable charge applies to the ceiling of time \( t \), ensuring that even if a partial minute is used, it rounds up, ensuring a whole minute charge. This ensures that the calculated cost fairly covers the duration of call usage, upholding the billing policy.

Discontinuity

Discontinuity in a function means that there are abrupt jumps or breaks in the graph of the function. It indicates points where the function does not transition smoothly from one value to another. This can be visually identified in the graph as sharp, sudden jumps.
In our problem, the cost function \( C(t) = 0.12 + 0.08 \, \lceil t \rceil \) experiences discontinuities at every positive integer value of \( t \).
This is a result of the ceiling function, which makes the cost jump up by \( \$0.08 \) exactly at each whole number of minutes. As time progresses from just before an integer to the integer itself, the cost suddenly increases without taking every possible value in between.

Ceiling Function

The ceiling function, often denoted by \( \lceil t \rceil \), is a mathematical function that rounds a number up to the nearest integer. It is especially useful in scenarios where units need to account for complete intervals, like in billing and pricing policies.
In our exercise about phone call billing, \( \lceil t \rceil \) ensures that any call time is rounded up to account for the next whole minute. For instance, if a call lasts 2.1 minutes, the ceiling function rounds it up to 3 minutes, effectively adding the charge for a full additional minute.
This rounding behavior can significantly impact cost calculation, leading to the characteristic step-like plot in the graph of our cost function, as each increment results in an additional charge.