Problem 61 - Algebra 1: Concepts and Skills

Question

The cost of a long-distance telephone call is \(\$.87\) for the first minute and \(\$.15\) for each additional minute. 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 costs for calls lasting 1, 2, 3, 4, 5, and 6 minutes based on the generated linear function are \$.87, \$1.02, \$1.17, \$1.32, \$1.47, and \$1.62 respectively.

Step 1: Formulate the cost equation.

The costs for the first minute are fixed at \(\$.87\). For each additional minute the cost increases by \(\$.15\). This can be written as: \(c = 0.87 + 0.15(t-1)\) for \(t > 1\). However, if \(t = 1\) then the cost \(c = 0.87\).

Step 2: Calculate the costs for different durations.

Based on the previous step, the cost of the calls that last 1, 2, 3, 4, 5 and 6 minutes can be calculated as follows:For \(t = 1\), \(c = 0.87\).For \(t = 2\), substituting into the formula, we obtain \(c = 0.87 + 0.15(2-1) = 1.02\).For \(t = 3\), \(c = 0.87 + 0.15(3-1) = 1.17\).For \(t = 4\), \(c = 0.87 + 0.15(4-1) = 1.32\).For \(t = 5\), \(c = 0.87 + 0.15(5-1) = 1.47\).And for \(t = 6\), \(c = 0.87 + 0.15(6-1) = 1.62\).

Step 3: Plot points for the costs.

Now that we've computed the costs for each given minute, these individual points can be plotted on a graph. The vertical axis would represent the total cost ‘c' and the horizontal axis would represent the time ‘t’. Each point (t,c) would then correspond to the cost of a call lasting 't' minutes.

Key Concepts

Understanding Linear Equations

When we talk about linear equations in the context of algebra, we're referring to an equation that makes a straight line when we graph it. These equations can be as simple as \( y = mx + b \) where \( m \) represents the slope of the line and \( b \) is the y-intercept, where the line crosses the vertical axis.

But how does this relate to our exercise? Here, we have a practical example with the equation \( c = 0.87 + 0.15(t-1) \) for \( t > 1 \) and \( c = 0.87 \) for \( t = 1 \) which represents the cost \( c \) of a long-distance call based on the duration \( t \) in minutes. Notice that this fits into our linear equation model, with the initial cost acting as the y-intercept and the rate per minute as the slope.

Breaking Down the Long-Distance Call Cost Equation

In our real-world equation, \( 0.87 \) is the base cost of a call (akin to \( b \) in the \( y = mx + b \) equation), and \( 0.15 \) represents the additional cost for each minute after the first, which is like the slope \( m \) since it shows how rapidly the cost rises per minute. To visualize the equation's linearity, one can use a graph, plotting the cost as the call duration increases.

Plotting Points on a Graph

Plotting points is a fundamental skill in algebra that translates equations into visual representations. By plotting points, we can see the relationship between variables in graphical form, making patterns more apparent and relationships easier to understand.

Step-by-Step Plotting

Let's apply this to the telephone call costs. For each minute \( t \) we calculated a corresponding cost \( c \) and these create pairs like (1, 0.87), (2, 1.02), (3, 1.17), and so on. Each of these pairs forms a point in two-dimensional space, with the first number (minutes) along the horizontal \( x \) axis, and the second number (costs) along the vertical \( y \) axis.

Once we have calculated these points, they're plotted on a graph. If the relationships are properly understood and the calculations are correct, they will form a straight line — confirming the equation's linearity. After plotting the points, you can draw a line through them to visually represent the cost equation.

Grasping Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a particular quantity or relationship without an equality sign. These expressions become equations once an equality is introduced, and they begin to describe a relationship between quantities.

An expression like \( 0.15(t-1) \) represents the additional amount one will pay for each minute after the first. In the context of our telephone call example, the expressions inside the equation showcase how the total cost \( c \) is built. The number \( 0.87 \) is a constant representing the initial cost, and \( 0.15(t-1) \) is a variable expression indicating the incremental cost per each additional minute after the first.

Connecting Expressions and Equations

Expressions are parts of an equation’s anatomy, and understanding them individually helps break down and solve more complex algebraic problems. In educational content, we often look at expressions in isolation to build students' understanding before moving on to full equations where these expressions interact to model situations like our telephone call scenario.