Problem 49 - Algebra 2

Question

To rent an inflatable trampoline for parties, it costs \(\$ 75\) an hour plus a set-up/tear-down fee of \(\$ 200 .\) Write an equation that represents this situation in slope-intercept form.

Step-by-Step Solution

Answer

The equation is \( y = 75x + 200 \).

Step 1: Identify known values

Start by identifying what is given in the problem. The problem states that it costs $75 per hour to rent the trampoline and there is a one-time setup/tear-down fee of $200.

Step 2: Define variables

Introduce variables to represent the quantities. Let \( x \) represent the number of hours the trampoline is rented, and \( y \) the total cost in dollars.

Step 3: Understand the equation components

The given situation can be translated to a linear equation, where the rate (cost per hour) is the slope, and the setup fee is the y-intercept.

Step 4: Write the slope-intercept equation

The general form of a slope-intercept equation is \( y = mx + b \). Here, \( m \) is the cost per hour (\\(75), and \( b \) is the fixed fee (\\)200). Substitute these values into the equation to get \[ y = 75x + 200 \].

Step 5: Verify the equation

Ensure that the equation represents the given situation. The term \( 75x \) ensures that for each hour rented, the cost increases by $75. The constant \( 200 \) confirms the one-time setup fee is added to the total cost.

Key Concepts

Slope-Intercept Form

The slope-intercept form is an essential concept in linear equations. When we talk about lines on a graph, the slope-intercept form helps us understand how those lines behave and interact.

The equation of a line in slope-intercept form is written as: \[ y = mx + b \]In this equation:

• **\(m\)** represents the slope of the line. It tells us how steep the line is and the direction it slopes—increasing or decreasing.
• **\(b\)** represents the y-intercept, which is the point where the line crosses the y-axis.
• **\(x\)** and **\(y\)** are the variables, with **\(x\)** usually representing the independent variable and **\(y\)** the dependent variable.

For instance, in the trampoline rental scenario, we have used this form to capture the cost dynamics. The cost per hour, 75, acts as the slope and the initial one-time fee of 200 is the y-intercept.

Variables in Algebra

Variables in algebra are symbols used to represent numbers or values that can change. They are fundamental in creating equations and expressions because they allow us to generalize problems and manipulate numerical relationships.

In our exercise, we defined:

• \(x\) as the number of hours the trampoline is rented. It's our independent variable, meaning the total cost depends on how many hours we rent the trampoline.
• \(y\) as the total rental cost. It's the dependent variable because it changes based on the number of hours.

Using variables allows flexibility in mathematical expressions and can simplify complex problems. By replacing unknowns with symbols, we can solve the equation to find specific values, such as how much it would cost for a particular rental duration.

Cost Analysis

Cost analysis involves breaking down expenses to understand the total cost structure. By examining the details of each cost component, we can make informed decisions and optimize spending.

With the trampoline rental problem, the cost analysis included two main components:

• **Per-hour rental fee (Variable Cost):** Each hour the trampoline is rented costs $75. This cost varies depending on how long the trampoline is used, making it a variable expense.
• **Setup/Takedown fee (Fixed Cost):** There is a one-time fee of $200. This fee stays constant regardless of the number of hours the trampoline is rented. It represents a fixed cost.

Together, these elements form a linear equation in the slope-intercept form, which represents the total expense as a function of time. Knowing how costs are structured allows us to budget accordingly and possibly adjust our hours of use to fit a specific budget.