Problem 5 - 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 Write equations that represent the cost of printing digital photos at each lab.

Step-by-Step Solution

Answer

EZ Online: \( C_\text{EZ}(x) = 0.15x + 2.70 \); Local Pharmacy: \( C_\text{Local}(x) = 0.25x \).

Step 1: Understanding the Problem

We need to determine the total cost of printing digital photos from two different options: EZ Online Digital Photos and Local Pharmacy. These costs include the price per photo and any additional charges, like shipping.

Step 2: Identify the Variables

Let \( x \) represent the number of digital photos printed. The cost functions will be based on this variable \( x \).

Step 3: Equation for EZ Online Digital Photos

The cost of printing at EZ Online Digital Photos is \( \\(0.15 \) per photo plus a \( \\)2.70 \) shipping charge. Therefore, the cost equation is \( C_\text{EZ}(x) = 0.15x + 2.70 \).

Step 4: Equation for Local Pharmacy

The cost of printing at the Local Pharmacy is \( \$0.25 \) per photo, with no additional shipping charges. Therefore, the cost equation is \( C_\text{Local}(x) = 0.25x \).

Key Concepts

Cost Functions

Cost functions allow us to represent costs mathematically. They give us a way to calculate the total cost of a service or product based on certain variables, such as the quantity of items or additional fees.
In this exercise, the cost functions reflect the total expense of printing a certain number of digital photos at two different places. For EZ Online Digital Photos, the cost function includes:

• A fixed shipping charge of \(2.70 that applies no matter how many photos you print.
• A variable cost of \)0.15 per photo, which will change based on the number of photos printed. This is because the more photos you print, the higher the total cost becomes due to this unit rate.

Thus, we have the cost function for EZ Online as:
\[ C_\text{EZ}(x) = 0.15x + 2.70 \]
Here, \(x\) is the number of digital photos. Every additional photo increases the cost by \(0.15.For Local Pharmacy, the cost function is simpler because it includes only a per-photo cost of \)0.25 and no extra charges. Therefore, the cost function is:
\[ C_\text{Local}(x) = 0.25x \]
This function shows that the total cost purely depends on the number of photos printed, as it multiplies $0.25 by the number of photos \(x\).

Variables in Equations

Variables in equations are symbols that stand for unknown values. In our problem, the variable \(x\) represents the number of digital photos printed. This variable is crucial because it allows the equations to calculate costs for any number of photos.
Why introduce variables like \(x\)? They make equations flexible. Instead of writing an equation for each possible number of photos, we use \(x\) to cover all cases. Whether you want to print 1 photo or 100 photos, using \(x\) allows us to plug that number directly into our cost functions.
The equations:

• \[ C_{EZ}(x) = 0.15x + 2.70 \]
• \[ C_{Local}(x) = 0.25x \]

You can substitute any integer for \(x\) to determine the cost for that specific number of photos. For example, if you wanted to calculate the cost for 10 photos, you would substitute 10 for \(x\). This adaptability is what makes variables powerful in equations.

Linear Equations

Linear equations are equations of the first order. When graphed, they produce a straight line. They typically look like \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In this exercise, both cost functions (\( C_{EZ}(x) \) and \( C_{Local}(x) \)) are linear equations. For the equation \( C_{EZ}(x) = 0.15x + 2.70 \), the slope \(m\) is 0.15, indicating how much the cost increases per additional photo. The y-intercept \(b\) is 2.70, representing the fixed shipping charge. This tells us that even if no photos are printed, there is an initial cost (fixed cost; in this case, the shipping fee).
On the other hand, the \( C_{Local}(x) = 0.25x \) equation is also linear but with \(b = 0\) indicating no initial fixed cost. Here, the slope \(m\) of 0.25 reflects the charge per photo. Linear equations help us model real-world scenarios in a straightforward way. By understanding how the lines graphically represent costs, we can easily visualize how costs change as more photos are printed.