Problem 4 - Algebra 1: Concepts and Skills

Question

You are going camping. The cost for renting a cabin at Shady Knoll Campground is \(\$ 65.00\) plus \(\$ 12.00\) per person. The cost in dollars is \(C=65+12 n,\) where \(n\) is the number of people. Copy and complete the input-output table. $$ \begin{array}{|ll|l|l|l|l|l|} \hline \text { Input } N & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Output } C & ? & ? & ? & ? & ? & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Here is the completed input-output table:\[\begin{array}{|l|l|l|l|l|l|l|}\hline \text { Input } N & 1 & 2 & 3 & 4 & 5 & 6 \\hline \text { Output } C & 77 & 89 & 101 & 113 & 125 & 137 \\hline\end{array}\]

Step 1: Understand the equation

The provided equation for the cost of renting a cabin is \(C=65+12n\), where \(n\) is the number of people. This means that the cost (\(C\)) equals to the base cost of renting the cabin (which is $65), plus 12 multiplied by the number of people (\(n\)).

Step 2: Apply the equation for each person count

Using the equation, calculate the cost for number of people ranging from 1 to 6. This is achieved by substituting the value of \(n\) in the equation \(C = 65 + 12n\). For instance, for 1 person, substituting \(n = 1\) in the equation gives \(C = 65 + 12*1 = 77\), which will be the output corresponding to \(n = 1\). Repeat this process for \(n\) from 2 to 6.

Step 3: Complete the table

Fill the output values calculated in step 2 in the input-output table under the Output \(C\) section corresponding to each Input \(N\). For example, for Input \(N\) as 1, fill the Output \(C\) as 77, for \(N = 2\), \(C = 89\), and so on.

Key Concepts

Linear Equations

Linear equations are foundational to algebra and are used to represent real-world situations where there is a constant rate of change. In our cabin rental algebra problem, the given equation, C = 65 + 12n, is a linear equation because it describes a relationship where the cost C increases by a constant amount as the number of people n increases.

The equation can be recognized as linear because it has two variables, C and n, and each variable is to the first power (e.g., no squares, cubes, etc.). Furthermore, the structure y = mx + b is evident here, with m being the constant rate (in this case, 12) and b the initial value or y-intercept (here, 65).

When visualized on a graph, this equation would form a straight line with a slope of 12, representing the rate at which costs increase per person, and a y-intercept at 65, indicating the base cost without any people.

Input-Output Tables

Input-output tables are a way of organizing information so that the relationship between variables can be easily understood. They are particularly handy when dealing with functions or equations in algebra.

To apply this concept to our cabin rental problem, you start with the input, which is the number of people n, and then use the algebraic equation to find the output, which is the cost C. The table helps you see at a glance how the cost changes as more people are added.

After calculating the cost for each number of people using the given equation, you fill in the 'Output C' row with these calculated costs. This simple visual organization of the data can be particularly beneficial for spotting patterns, predicting future outcomes, or even checking the results of calculations for consistency.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific quantity. In the context of our camp rental problem, C = 65 + 12n is the algebraic expression that represents the total cost C of renting a cabin based on the number of people n.

This expression includes the constant 65, which is the flat fee, and the variable term 12n, where 12 is the rate per person. Understanding how to manipulate these expressions is crucial in algebra. For example, if you know there are 5 people, substituting n with 5 results in C = 65 + 12(5), or C = 125. This calculation shows how algebraic expressions are practical tools for solving real-world mathematical problems by plugging in different values for variables.