Let's break down the problem presented in the image.

Part A: Write the function model represented by the data in the table.

The table gives us the relationship between the amount of juice made and the concentrate remaining:

• Juice Made (ounces): 200, 600, 1,000, 1,400
• Concentrate Remaining (ounces): 280, 200, 120, 40

We can model this relationship as a linear equation, since the relationship between juice made and concentrate remaining appears to decrease at a constant rate. Let:

• $y$ = Concentrate remaining
• $x$ = Juice made

We need to find the slope ($m$) and y-intercept ($b$) to write the equation in the form $y = mx + b$.

To find the slope: $m = \frac{\Delta y}{\Delta x} = \frac{200 - 280}{600 - 200} = \frac{-80}{400} = -0.2$ Thus, the slope $m = -0.2$.

Now, using one point from the table to solve for $b$, we can plug in the values $(x = 200, y = 280)$ into the equation $y = mx + b$: $280 = -0.2(200) + b$ $280 = -40 + b$ $b = 320$

Thus, the equation of the line is: $y = -0.2x + 320$

Part B: How many ounces of concentrate were in the original container before any juice was made?

From the equation $y = -0.2x + 320$, we can find the amount of concentrate when no juice was made by substituting $x = 0$: $y = -0.2(0) + 320 = 320$ So, there were 320 ounces of concentrate in the original container.

Part C: How many ounces of juice can be made using the whole container of concentrate?

When the entire container of concentrate is used up, the concentrate remaining will be zero. We can find the amount of juice made when $y = 0$: $0 = -0.2x + 320$ $0.2x = 320$ $x = \frac{320}{0.2} = 1,600$ So, 1,600 ounces of juice can be made using the whole container of concentrate.

Would you like any further clarifications or details on the steps?

Here are five related questions to expand on this problem:

1. How would you interpret the slope of the function in real-world terms?
2. Can you explain how to find the y-intercept from the table values?
3. What would happen to the function if the juice-making process became more efficient?
4. How can you verify that the relationship is linear based on the table values?
5. What would the graph of this linear relationship look like?

Tip: When identifying a linear relationship, always check if the rate of change between points (the slope) is consistent across the table values.