Use your notes from Friday 10/15  Problem solving with rate of change and initial value. It is similar to the Strawberry farm question. 



A 2 ounce box of Cheez-Its costs $5. A 4 ounce box costs $9. A third box of Cheez-Its costs $15. 



A) Use the given relationship to write a linear equation that expresses the price of a box of Cheez-Its as a function of the number of ounces of Cheez-Its in the box? 
 

B) Calculate the number of ounces in the third box of Cheez-its.  
 ounces

To solve the problem regarding the cost of Cheez-Its, we will break it down step by step.

### A) Writing the Linear Equation

We are given the following data points regarding the price (P) of Cheez-Its in relation to their weight in ounces (x):

1. For 2 ounces: $ P = 5 $
2. For 4 ounces: $ P = 9 $

We can use these points to find the slope (rate of change) and the initial value (y-intercept) for the linear equation $ P = mx + b $.

#### Step 1: Calculate the Slope (m)

The slope $ m $ can be calculated using the formula:

\[
m = \frac{P_2 - P_1}{x_2 - x_1}
\]

Using the points (2, 5) and (4, 9):

\[
m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2
\]

#### Step 2: Use the Slope and a Point to Find the Y-Intercept (b)

Using the slope $ m = 2 $ and one of the points (e.g., (2, 5)), we can substitute into the equation to find $ b $:

\[
5 = 2(2) + b
\]

\[
5 = 4 + b \implies b = 1
\]

#### Step 3: Write the Equation

Now, we can write the linear equation:

\[
P = 2x + 1
\]

### B) Calculate the Number of Ounces in the Third Box

We know the price of the third box is $15. We can use our linear equation to find the number of ounces (x):

\[
15 = 2x + 1
\]

Subtract 1 from both sides:

\[
14 = 2x
\]

Now, divide by 2:

\[
x = 7
\]

### Summary of Results

- **A)** The linear equation expressing the price as a function of ounces is $ P = 2x + 1 $.
- **B)** The number of ounces in the third box of Cheez-Its is **7 ounces**.

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the equation change if the price of the 4-ounce box increased?
2. What would be the cost of a 6-ounce box based on this model?
3. How can we interpret the slope in the context of this problem?
4. If we plotted this on a graph, what would the x-intercept represent?
5. How would the initial price affect the overall price per ounce?

**Tip:** When analyzing linear relationships, always identify the slope and intercept first to understand the behavior of the function.

Use your notes from Friday 10/15 Problem solving with rate of change and initial value. It is similar to the Strawberry farm question. A 2 ounce box of Cheez-Its costs $5. A 4 ounce box costs $9. A third box of Cheez-Its costs $15. A) Use the given relationship to write a linear equation that expresses the price of a box of Cheez-Its as a function of the number of ounces of Cheez-Its in the box? B) Calculate the number of ounces in the third box of Cheez-its.

Explore how to derive the price function for Cheez-Its based on ounces. Given that a 2-ounce box costs $5 and a 4-ounce box costs $9, we calculate the linear equation and find the ounces in a $15 box, revealing the relationship between weight and cost. Suitable for Grades 7-9.

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