Problem 6 - Algebra 2

Question

Developing Digital Photos EZ Online Digital Photos Charges \(\$ 0.15\) per digital photo and \(\$ 2.70\) for shipping Local Pharmacy Charges \(\$ 0.25\) per digital photo Under what conditions is the cost to print digital photos the same for either store?

Step-by-Step Solution

Answer

For 27 photos, the cost is the same at both EZ Online and the Local Pharmacy.

Step 1: Define the cost functions

First, we must define the cost for each service. For EZ Online Digital Photos, the cost function is comprised of a per-photo cost and a fixed shipping cost. Let's represent the number of photos as \( n \). The cost function is given by:\[ C_{EZ}(n) = 0.15n + 2.70 \]For the Local Pharmacy, there's only a per-photo charge, so the cost function is:\[ C_{LP}(n) = 0.25n \]

Step 2: Set the cost equations equal

Next, to find the point where the costs are equal, we need to set the two cost functions equal to each other:\[ 0.15n + 2.70 = 0.25n \]

Step 3: Solve for the number of photos

Now, let's solve for \( n \):1. Subtract \( 0.15n \) from both sides to move all variables to one side of the equation:\[ 2.70 = 0.25n - 0.15n \]2. Simplify:\[ 2.70 = 0.10n \]3. Divide both sides by \( 0.10 \) to isolate \( n \):\[ n = \frac{2.70}{0.10} = 27 \]

Step 4: Verify the solution

To ensure the solution is correct, substitute \( n = 27 \) back into both cost equations:1. \( C_{EZ}(27) = 0.15 \times 27 + 2.70 = 4.05 + 2.70 = 6.75 \)2. \( C_{LP}(27) = 0.25 \times 27 = 6.75 \)Both costs are equal, verifying that \( n = 27 \) is indeed the solution.

Key Concepts

Solving Equations

Understanding how to solve equations is a fundamental mathematical skill. It involves finding the values of variables that satisfy given mathematical statements. In the exercise, two different cost equations are given, each representing the costs associated with printing digital photos through two different services. By setting these equations equal to each other, we seek to find the number of photos, denoted as \( n \), at which both services would charge the same amount.

The first step in solving such equations is to arrange the terms of the equation so that all terms involving the variable are on one side, and constant terms are on the other. In this case, we start with the equation \( 0.15n + 2.70 = 0.25n \). We subtract \( 0.15n \) from both sides to simplify.

The equation becomes \( 2.70 = 0.10n \), which simplifies further to \( n = \frac{2.70}{0.10} \), resulting in \( n = 27 \). This process is a straightforward example of how combining like terms, using basic operations, and isolating the variable can help solve equations effectively.

Digital Photo Printing Costs

Digital photo printing costs can vary depending on the service provider. Here, two services are compared: EZ Online Digital Photos and a Local Pharmacy. Each has a distinct pricing structure, influencing your total costs depending on the number of photos you print.

EZ Online Digital Photos charges \( \\(0.15 \) per photo plus a fixed shipping cost of \( \\)2.70 \). In contrast, the Local Pharmacy solely charges \( \$0.25 \) per photo without any additional fees. Understanding these cost structures allows customers to make informed decisions about which service might be more economical based on the quantity of photos they need to print.

Economies of scale play a crucial role here. With variable costs depending on quantity, more extensive orders through EZ Online Digital Photos might be more economical after covering the initial shipping charge, compared to Local Pharmacy's flat per-photo rate. Exploring which option is cheaper for different photo counts can demonstrate cost-effectiveness and budgeting strategies.

Mathematical Modeling

Mathematical modeling involves using mathematical concepts to represent real-world scenarios. In this exercise, we model the cost of printing digital photos with different services using equations that reflect each service's pricing structure.

This model is an example of how mathematical reasoning translates qualitative observations into quantitative analysis. By defining cost functions, \( C_{EZ}(n) = 0.15n + 2.70 \) and \( C_{LP}(n) = 0.25n \), we convert service details into an algebraic framework.

This structured approach allows us to compare costs effectively and find solutions, like when both costs are equal, by setting and solving the equations. This demonstrates one of modeling's key purposes: simplifying complex systems to make predictions or comparisons.

• Establishes clear decision-making frameworks.
• Highlights the importance of isolating variables and constants.
• Emphasizes cost-benefit analysis through algebraic methods.

Mathematical modeling is vital not only in economics but also in various fields where quantitative problem evaluation is necessary.