Problem 80 - Algebra 1

Question

Use the following information. A pizzeria charges \(\$ 6.00\) for a large cheese pizza, and \(\$ .85\) for each additional topping. The total cost \(C\) of a large cheese pizza with \(n\) additional toppings is given by \(C=6+0.85 n . Write an input-output table that shows the total cost of a pizza with \)0,1,2,3$ 4, and 5 additional toppings.

Step-by-Step Solution

Answer

The input-output table is: n = 0, C = $6.00; n = 1, C = $6.85; n = 2, C = $7.70; n = 3, C = $8.55; n = 4, C = $9.40; n = 5, C = $10.25.

Step 1. Understand the Given Function

The given function \(C = 6 + 0.85n\) represents the cost of a pizza. Here, \(C\) is the total cost and \(n\) is the number of additional toppings. We understand that a base cost of \(\$6.00\) is charged for a large cheese pizza and an additional charge of \(\$0.85\) is added for each topping.

Step 2: Applying Function to Values

Apply the function to each given value of \(n\) (0,1,2,3,4,5). Use the formula \(C = 6 + 0.85n\) for each value of \(n\). For example, for \(n = 0\), \(C = 6 + 0.85*0 = \$6.00\). This means a pizza with no additional toppings costs \$6.00.

Step 3: Tabulate the Results

Record the computed total cost \(C\) for each value of \(n\) (0,1,2,3,4,5) in a table. The first column of the table should be 'Number of Toppings (n)', and the second 'Total Cost (C)'.

Key Concepts

Cost Function

A cost function in mathematics and economics is a formula used to determine how much it costs to produce or buy something. In our pizzeria example, the cost function is represented by the equation: \[C = 6 + 0.85n\], where:

• \(C\) is the total cost of the pizza.
• The term \(6\) represents the fixed base cost, which remains the same no matter how many toppings you add. This is the cost of a large cheese pizza without any additional toppings at \(\\(6.00\).
• \(0.85n\) indicates the variable cost, which depends on the number \(n\) of additional toppings.

So, the cost function helps us find out how the total cost \(C\) changes as we change the number of toppings \(n\). This is an example of a linear function, where the cost increases by a constant rate of \(\\)0.85\) per topping.

Input-Output Table

An input-output table is a simple way to understand how changing one quantity (input) affects another (output). In our case:

• The input is the number of additional toppings \(n\).
• The output is the total cost \(C\).

By creating a table, we can visually represent how these two quantities interact as you increase the number of toppings from 0 to 5:

• For 0 toppings: \(C = 6 + 0.85 \times 0 = \\(6.00\)
• For 1 topping: \(C = 6 + 0.85 \times 1 = \\)6.85\)
• For 2 toppings: \(C = 6 + 0.85 \times 2 = \\(7.70\)
• For 3 toppings: \(C = 6 + 0.85 \times 3 = \\)8.55\)
• For 4 toppings: \(C = 6 + 0.85 \times 4 = \\(9.40\)
• For 5 toppings: \(C = 6 + 0.85 \times 5 = \\)10.25\)

This table makes it clearer how the cost changes with more toppings, which is especially useful for quick references.

Algebraic Expression

An algebraic expression consists of numbers, variables, and operations (like addition or multiplication). It is used to describe mathematical ideas and relationships. In our pizzeria scenario, our algebraic expression is:\[C = 6 + 0.85n\].Here's a breakdown of each part:

• \(6\): The base cost of the pizza, which is constant. This value is not affected by the toppings.
• \(0.85\): This is the coefficient that describes how much each additional topping adds to the base price.
• \(n\): The variable, representing the number of additional toppings.

Together, these components make the algebraic expression a powerful tool to calculate costs quickly for different amounts of toppings. It shows a direct and simple relationship, which makes it easy to use for calculations or predictions.