Problem 37 - Algebra and Trigonometry

Question

Cost of a Pizza \(\quad\) A pizza parlor charges \(\$ 7.95\) for a cheese pizza and \(\$ 1.25\) for each topping. (a) How much does a 3 -topping pizza cost? (b) Find a formula that models the cost \(C\) of a pizza with \(n\) toppings. (c) If a pizza costs \(\$ 14.20,\) how many toppings does it have?

Step-by-Step Solution

Answer

(a) $11.70 (b) C(n) = 7.95 + 1.25n (c) 5 toppings

Step 1: Calculate Cost of a 3-topping Pizza

To find the cost of a pizza with three toppings, start with the base price of a cheese pizza, which is \(7.95. Add the cost of three toppings, where each topping costs \)1.25. Calculate the total cost: \[7.95 + 3 \times 1.25 = 7.95 + 3.75 = 11.70\]So, the cost of a 3-topping pizza is $11.70.

Step 2: Derive Formula for Pizza Cost

Model the cost \( C(n) \) of a pizza with \( n \) toppings. The formula is the sum of the constant base price and the variable cost dependent on the number of toppings. This results in:\[C(n) = 7.95 + 1.25n\]

Step 3: Determine Number of Toppings for Specific Cost

Given that a pizza costs $14.20, use the formula from Step 2 to find the number of toppings \( n \). Set \( C(n) = 14.20 \) and solve for \( n \):\[14.20 = 7.95 + 1.25n\]Subtract 7.95 from both sides:\[14.20 - 7.95 = 1.25n\]\[6.25 = 1.25n\]Divide both sides by 1.25:\[n = \frac{6.25}{1.25} = 5\]Thus, the pizza has 5 toppings.

Key Concepts

Mathematical Modeling

In mathematics, modeling provides a way to represent real-world situations with mathematical equations. It helps us visualize and solve problems more effectively. For example, when considering the cost of a pizza with various toppings, mathematical modeling allows us to construct an equation that reflects the relationship between the number of toppings and the total price.

To create a model, we break down the problem into manageable components. In our case, the components are the base cost of the pizza and the cost per topping. The formula we derived is:

• Base cost of a cheese pizza: \(7.95
• Cost per topping: \)1.25

We combine these parts to form the model
\[ C(n) = 7.95 + 1.25n \], where

• \( C(n) \) is the cost of the pizza,
• \( n \) is the number of toppings.

This formula, or mathematical model, captures the interaction between price and toppings, allowing for predictions and calculations.

Linear Equations

Linear equations form the backbone of many algebra problems, as they describe a straight-line relationship between variables. They are simple yet powerful tools in mathematics. In the context of our pizza problem, the equation \[ C(n) = 7.95 + 1.25n \] is a linear equation.

A linear equation typically has the form:
\[ y = mx + b \], where

• \( m \) represents the slope,
• \( b \) represents the y-intercept.

In our pizza equation:

• \( m = 1.25 \) since this coefficient represents the additional cost for each topping,
• \( b = 7.95 \) is the fixed cost of the cheese pizza.

This structure illustrates the linear increase in cost as more toppings are added. Understanding linear equations helps us solve for unknowns and appreciate their simplicity in modeling consistent relationships.

Problem Solving

Problem solving in mathematics involves using various strategies to find a solution. It encourages logical thinking and perseverance. With our pizza cost scenario, we are tasked with answering specific questions through arithmetic and algebraic reasoning.

The problem is broken into smaller, manageable steps:

• Firstly, calculate the cost of a 3-topping pizza using the formula: \[ 7.95 + 3 \times 1.25 = 11.70 \].
• Secondly, derive a general formula, \( C(n) = 7.95 + 1.25n \), to determine the cost for any number of toppings.
• Finally, use the equation to find the number of toppings when the total cost is given: \( 14.20 = 7.95 + 1.25n \).

Each step requires us to apply different skills, like substitution and solving equations. By practicing the problem solving method systematically, students enhance their ability to tackle a wide array of mathematical challenges.