Problem 64 - College Algebra with Corequisite Support

Question

If we rent a truck and pay a \(\$ 75 /\) day fee plus \(\$ .20\) for every mile we travel, write a linear equation that would express the total cost per day \(y,\) using \(x\) to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.

Step-by-Step Solution

Answer

The total cost for 70 miles is \$89.

Step 1: Understand the Problem Context

The cost consists of a daily fee and a per-mile charge. You're given a fixed daily fee of $75 and an additional charge of $0.20 per mile traveled.

Step 2: Formulate the Linear Equation

The total cost per day, represented by \( y \), can be expressed as the sum of the fixed daily fee and the variable cost per mile. Therefore, the cost equation is: \( y = 75 + 0.20x \), where \( x \) is the number of miles traveled.

Step 3: Graph the Linear Equation

To graph the function \( y = 75 + 0.20x \), plot \( y \) on the vertical axis and \( x \) on the horizontal axis. Start at the point \((0, 75)\), which is the y-intercept, and use the slope of 0.20 to determine another point, e.g., at \( x = 10 \), \( y = 77 \). Connect these points with a straight line.

Step 4: Calculate the Total Cost for 70 Miles

Substitute \( x = 70 \) into the equation \( y = 75 + 0.20x \). Calculate: \( y = 75 + 0.20 \times 70 = 75 + 14 = 89 \). The total cost for traveling 70 miles is \$89.

Key Concepts

Cost Function

A cost function is a mathematical expression that calculates how much you have to pay based on certain conditions. In the context of renting a truck, the cost function includes both a fixed fee and a variable fee. The fixed fee is what you pay no matter what—just like a subscription service that costs the same every month. In our example, the fixed fee is \(75 per day. This means even if you don't drive a single mile, you still have to pay \)75.

The variable part of the cost is the \(0.20 you pay for each mile driven. So, if you drive more miles, this part of your cost will increase. We combine these two fees in a linear equation:

• The fixed daily fee: \)75
• The per-mile charge: $0.20 per mile
• The cost function: \( y = 75 + 0.20x \), where \( x \) is the number of miles

This equation helps you calculate how much you will spend depending on the miles you drive.

Graphing Linear Equations

Graphing linear equations can make the relationships in your data much clearer. To display our cost function graphically, follow these steps:

• Identify the y-intercept, where the line crosses the vertical y-axis. In our equation \( y = 75 + 0.20x \), the y-intercept is 75, representing the cost when no miles are driven.
• The slope of the line reveals how much the cost increases per additional mile. Here, the slope is 0.20, meaning each mile adds \(0.20 to the cost.
• Start plotting by marking the y-intercept at (0, 75) on your graph.
• Using the slope of 0.20, find another point by moving right one unit (mile) and up 0.20 units (dollars). For example, after 10 miles, the cost is \)77, so plot the point (10, 77).
• Connect these points with a straight line extending across the graph.

The line you've drawn shows you visually how costs change with miles driven, making it easier to estimate, plan, and understand your expenses.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so their shape is clear at a glance. It is usually written as \( y = mx + b \), where:

• \( y \) is the total cost or dependent variable.
• \( x \) is the independent variable, in this case, the miles driven.
• \( m \) represents the slope. It is the rate of change, showing how much \( y \) changes for every increase in \( x \). In our example, the slope is 0.20, indicating an incremental cost of \(0.20 per mile.
• \( b \) is the y-intercept. It is the value of \( y \) when \( x \) is zero. Here, \( b = 75 \), which means even with zero miles driven, the cost is \)75.

When using the slope-intercept form, you can easily identify both the starting point of an equation on a graph (the y-intercept) and how steep the line is (the slope). This form not only aids in graphing but also in understanding how different parts of the equation contribute to the overall cost.