Problem 14 - Algebra 1: Concepts and Skills

Question

In Exercises \(12-17\), use the following information. Renting a canoe costs 10 dollars plus 28 dollars per day. The linear model for this situation relates the total cost of renting a canoe, \(y,\) with the number of days rented, \(x\). Use the slope and \(y\) -intercept form to write the linear model.

Step-by-Step Solution

Answer

The linear model for the given situation is \(y = 28x + 10\).

Step 1: Identify Slope and Y-intercept

The problem gives us that it costs $10 initially to rent a canoe, and an additional $28 per day. This means that the slope (m), or the rate at which the total cost increases each day, is $28. The y-intercept (c), or the initial cost of renting a canoe, is $10.

Step 2: Insert Slope and Y-intercept in the linear equation

Now that we have identified the slope and y-intercept, we will insert them into the slope-intercept form of a linear equation, which gives us the equation \(y = 28x + 10\) where 'y' represents the total cost and 'x' stands for the number of days.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations where you can easily identify two important components: the slope and the y-intercept. The standard form of this equation is \(y = mx + c\). Here:

• \(y\) stands for the dependent variable, which is often the outcome or result we want to find. In our context, it is the total cost of renting the canoe.
• \(x\) is the independent variable, usually representing the input or what we control or decide. Here, it refers to the number of days the canoe is rented.
• \(m\) symbolizes the slope of the line, showing how much \(y\) will change when \(x\) changes.
• \(c\) is the y-intercept, indicating where the line crosses the y-axis when \(x = 0\).

Knowing this form helps us quickly understand the relationship between variables and allows for easy graphing. This form is especially useful for visualizing how any increase in the number of days rented affects the total cost.

Slope

The slope, represented as \(m\) in the equation, describes the rate of change between the variables. In the context of rental costs:

• The slope \(m = 28\) tells us that the total cost increases by 28 dollars for every additional day the canoe is rented. This is because each day adds this amount to the initial cost.
• The concept of slope is crucial as it provides insight into how steeply or gradually the cost rises over time.
• A larger slope value means a quicker increase in the overall cost, while a smaller slope would indicate a slower rise.

Understanding this "rate of change" is vital for budgeting and planning, as it clarifies how the decision of how many days to rent can impact the total expense. If you ever graph the equation, you'll see that the slope determines the angle or steepness of the line on the graph.

Y-Intercept

The y-intercept, indicated as \(c\) in the slope-intercept form, is the starting point on a graph where the line crosses the y-axis. In this canoe rental example:

• The y-intercept \(c = 10\) represents the initial cost of renting the canoe, before any additional days are considered.
• This means that even if you rent the canoe for zero days, you will still incur a cost of 10 dollars.
• It provides a clear base value from which the total cost calculation begins.

The y-intercept is significant because it provides the baseline from which all changes are measured as the number of rental days increases. In practical terms, knowing this helps you understand fixed costs that will occur regardless of usage time.