Problem 12 - 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, \(x\). What number corresponds to the slope in the linear model?

Step-by-Step Solution

Answer

The number that corresponds to the slope in the linear model is 28 dollars per day.

Step 1: Identify the linear model

The problem provides that the renting of a canoe follows a linear model, meaning the total cost, \(y\), is a rectangle linear function of the number of days, \(x\). The cost function for this problem is \(y = 28x + 10\).

Step 2: Identify the slope

In the linear model, the coefficient of \(x\), which in this case is 28, is the slope of the function. The slope represents the increase in total cost for every additional day the canoe is rented.

Key Concepts

Slope

When working with linear equations, the slope is a key component that tells us how the relationship changes between the two variables. In our canoe rental scenario, the slope is represented by the rate of $28 per day. This means for each additional day you rent a canoe, your total cost increases by 28 dollars.
To put it simply, the slope in a linear model is the coefficient of the dependent variable, which shows how much one unit change in
the independent variable influences the dependent variable. In mathematical terms, if our equation is written as \( y = mx + b \), "m" is the slope.
The slope is vital as it informs you about the steepness and direction of the line on a graph. A positive slope indicates an upward trend in cost as days increase, while a negative slope would suggest a decrease, which isn't the case here.

Cost Function

A cost function provides a way to calculate the total expense associated with a certain level of activity. It's an application of linear equations broadly used across various fields, particularly in economics and business.
In our example, the cost function is given by \( y = 28x + 10 \). This function helps calculate the total cost \( y \) of renting a canoe for 'x' number of days.

• The first part, \( 28x \), represents the variable costs changing with days. For every day of rental, you incur 28 dollars.


• The second part, \( 10 \), is the fixed cost. It's the amount you pay regardless of how many or few days you rent the canoe.

Understanding the cost function is crucial to budgeting and predicting expenses, both in personal finance and larger business operations. It serves as a straightforward tool to forecast costs over varying levels of usage or production.

Linear Model

A linear model is a mathematical representation showcasing the relationship between two variables. It aims to predict or explain how changes in one variable affect another. With our canoe rental example, this relationship is represented by \( y = 28x + 10 \), where "\( y \)" is the total cost and "\( x \)" is the number of days rented.
Linear models have several defining characteristics:

• They show a constant rate of change, which remains uniform across the graph's entire line.


• The graph of the equation is a straight line, showcasing the direct proportionality between the dependent and independent variables.


• These models simplify complex real-life situations into basic, understandable terms, making them a foundational element of algebra.

Through linear models, we can break down costs, like the canoe rental, in simple terms, allowing individuals and businesses to make informed decisions based on predictable outcomes.