Problem 80 - College Algebra

Question

A cellphone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot$$\$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120\end{array}\right.$$ Plan \(B\) \(\cdot \$ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 200 \\\40+0.30(t-200) & \text { if } t>200\end{array}\right.$$ Simplify the algebraic expression in the second line of the piecewise function for plan \(\mathrm{B}\). Then use point-plotting to graph the function.

Step-by-Step Solution

Answer

The simplified expression for the second line of the piecewise function for plan B is \( -20 + 0.30t \) when \( t > 200 \). After plotting the points and creating the graph, we see that the function remains flat at \( C(t) = 40 \) for \( 0 \leq t \leq 200 \), then increases linearly with a slope of 0.30 for \( t > 200 \).

Step 1: Simplify the expression for Plan B

The algebraic expression for the second line of Plan B is \( 40 + 0.30*(t-200) \). We can simplify this by expanding the bracket, which gives us \( 40 + 0.30t - 60 \). After that, we combine the constants to simplify the equation further, resulting in the function \( -20 + 0.30t \) when \( t > 200 \). So, the simplified piecewise function for plan B becomes: \[ C(t) = \begin{cases} 40 & \text{if } 0 \leq t \leq 200 \ -20 + 0.30t & \text{if } t > 200 \end{cases}\]

Step 2: Graph the function for Plan B

To graph the function for plan B, we need to plot points for both the sections of the function. For the first part of the function, which is \( C(t) = 40 \) for \( 0 \leq t \leq 200 \), we plot two points at (0,40) and (200,40) and draw a straight line between them. For the second part of the function, which is \( C(t) = -20 + 0.30t \) for \( t > 200 \), we plug in 200 for \( t \) in the equation. This gives us \( -20 + 0.30*200 = 40 \), so we continue the graph from this point (200,40), with a slope of 0.30.

Step 3: Mark the coordinates and join the points

Mark the points on the graph and join each consecutive pair of points with a straight line, remembering to account for the break in the function. Make sure to establish a clear cut-off for \( C(t) = 40 \) at \( t = 200 \), differentiating the two separate equations of the piecewise function.

Key Concepts

Simplify Algebraic Expressions

Understanding how to simplify algebraic expressions is fundamental in studying piecewise functions. Simplification of expressions enables us to make calculations easier and to understand the relationship between variables more clearly. In the context of Plan B from the given exercise, the simplification process involves distributing multiplication over subtraction.

Starting with the unsimplified expression, \( 40 + 0.30 \times (t-200) \), our first step is to distribute the \( 0.30 \) across the terms within the parentheses. This gives us two separate terms: \( 0.30 \times t \) and \( 0.30 \times -200 \). Multiplying these out, we get \( 0.30t \) and \( -60 \), respectively. Combining these with the constant \( 40 \), we end up with \( 40 + 0.30t - 60 \), which then simplifies to \( -20 + 0.30t \).

When simplifying, remember to:

• Apply the distributive property correctly.
• Combine like terms, both constants, and those with the same variable,
• Check your work for any possible additional simplifications.

Simplifying expressions can also aid in error checking, as complex expressions often obscure mistakes.

Graphing Functions

Graphing is a powerful tool for visualizing the behavior of mathematical functions, like those given in the exercise. The ability to graph a function is invaluable when you want to quickly grasp how a particular plan behaves over time. For Plan B, we graph the piecewise function by considering each piece one at a time.

The constant piece \( C(t) = 40 \) for \( 0 \leq t \leq 200 \) is straightforward because it represents a flat line where the cost remains the same, irrespective of the minutes used, up to 200 minutes. Plotting this on a graph, we get a horizontal line from (0, 40) to (200, 40).

For the second piece where \( t > 200 \), the function is \( C(t) = -20 + 0.30t \), representing a linear equation that starts at 200 minutes and increases at the rate of \$0.30 for every additional minute. Since \( C(200) = 40 \), the graph continues from the point (200, 40) upwards and rightwards, matching the slope of 0.30.

To render this graph accurately, start by plotting the y-intercept and other strategic points, then draw a line connecting these points. Always ensure any transitions between slices of the piecewise function are clear on the graph.

Linear Equations

At the heart of piecewise functions, which include plans like B, are linear equations. These equations form straight lines when graphed, and their general form is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. Understanding how to work with linear equations helps you quickly construct a graph of a piecewise function or analyze its behavior.

For Plan B's second part, where \( t > 200 \), we have the linear equation \( -20 + 0.30t \). Here, 0.30 is the slope (\( m \)), showing the rate of change—how much the cost increases per additional minute. The constant term, -20, results from shifting the y-axis to match the context of the piecewise function.

Remember, in the context of linear equations:

• The slope (\( m \)) indicates the steepness and direction of a line.
• The y-intercept (\( b \)) provides the starting value on the vertical axis at \( t = 0 \).
• Since piecewise functions can change formulas at certain points, the concept of a 'piecewise' y-intercept might be applied, as is the case with the shift at \( t = 200 \) minutes.

Understanding these aspects will help you handle piecewise functions accurately during both simplification and graphing.