Problem 60 - Algebra 1

Question

the cost of a long-distance telephone call is \(\$ .87\) for the first minute and \(\$ .15\) for each additional minute. PLOTTING POINTS Let \(c\) represent the total cost of a call that lasts \(t\) minutes. Plot points for the costs of calls that last \(1,2,3,4,5,\) and 6 minutes.

Step-by-Step Solution

Answer

The plot of the call cost consists of the points (1, .87), (2, 1.02), (3, 1.17), (4, 1.32), (5, 1.47), (6, 1.62)

Step 1: Calculate the costs for each duration

First, calculate the cost of the call for durations of \(1, 2, 3, 4, 5, 6\) minutes. For the first minute it's \(\$ .87\). For each additional minute, add \(\$ .15\) to the previous total cost. So, for example, for a 2-minute call, the cost would be \(\$ .87 + \$ .15 = \$ 1.02\). Continue this calculation for each duration.

Step 2: Create the points

Using the costs calculated in the previous step, create points where the x-coordinate is the call duration (in minutes) and the y-coordinate is the cost. For example, the cost of a 1-minute call is \(\$ .87\), so the first point is (1, .87). Similarly, the cost of a 2-minute call is \(\$ 1.02\), so the second point is (2, 1.02). Continue this process for each call duration.

Step 3: Plot the points

Finally, plot the points on a graph. The x-axis represents the call duration in minutes and the y-axis represents the cost. Connect the points to form a line that represents the cost of a call as the function of its duration. This line shows how the cost increases as the call duration increases.

Key Concepts

Plotting Points

Plotting points is a foundational skill in mathematics, particularly when visualizing relationships in algebra. To plot points, you need a coordinate system, usually with horizontal (x-axis) and vertical (y-axis) lines. Each point is an ordered pair \( (x, y) \) representing its position on the grid.

For example, in our given exercise, the cost of a telephone call is related to its duration. The first step is to interpret the problem and determine the points to plot. The x-coordinate corresponds with the call duration \( t \) in minutes, and the y-coordinate corresponds with the total cost \( c \) of the call.

Using the calculated values from the step by step solution, we get points like \( (1, .87) \) for a 1-minute call and \( (2, 1.02) \) for a 2-minute call. To plot these points, start at the origin, move right to the x-coordinate value, and then move up or down to reach the y-coordinate value. The points for 3, 4, 5, and 6-minute calls would be found in the same way and placed on the graph accordingly.

Once all points are on the graph, they can be connected to form a visual representation of the cost function with respect to time.

Function Representation

Function representation refers to the different ways a function can be expressed. For instance, a function can be depicted algebraically, graphically, verbally, or in a table of values.

In the context of our exercise, the function representing the cost of a long-distance call is initially described verbally and then translated into a set of points. Algebraic representation would involve an equation, while graphical representation involves plotting the function on a graph.

To understand the concept, consider our exercise: for the first minute, the cost is a fixed value, and for each additional minute, a consistent incremental amount is added. This can be represented graphically by plotting the points calculated earlier. When connected, these points form a line on the graph, which clearly shows how the cost function behaves over time. If you drew a table, the left column could list call duration, and the right column would show corresponding costs.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a specific quantity. They are the building blocks for algebraic equations and functions.

In our exercise, the formula for calculating call costs is a real-world example of an algebraic expression. The fixed cost for the first minute is \( .87 \) dollars, and the cost for each additional minute is \( .15 \) dollars. To express the total cost \( c \) for \( t \) minutes, we would write \( c = .87 + (.15 \times (t - 1)) \) with the understanding that this applies for all calls longer than one minute.

In this expression, \( .15 \times (t - 1) \) accounts for the additional cost past the first minute, and \( .87 \) represents the base cost. Such expressions help us swiftly calculate the total for any call duration, emphasizing the practicality and importance of algebra in everyday situations.