Question

The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed. Write a function rule that models this situation
Numbers of Meales dispensed x:1,3,9,7
Amount of pet food remaining y: 21,15,6,3

Answer 1

To create a function rule that models the relationship between the number of meals dispensed (x) and the amount of pet food remaining (y), we can analyze the provided data:

- **Number of meals dispensed (x)**: 1, 3, 7, 9
- **Amount of pet food remaining (y)**: 21, 15, 3, 6

It looks like there is a pattern between the number of meals dispensed and the amount of pet food remaining.

1. **Calculate how much pet food is dispensed per meal**:
- From the first data point, when 1 meal is dispensed, there are 21 cups remaining. The change to the second point, when 3 meals are dispensed, yields:
\[
21 - 15 = 6 \text{ cups (dispensed for 2 meals)}
\]
Thus, for each meal, it appears that \(\frac{6}{2} = 3\) cups are dispensed.

2. **Check consistency with other points**:
- **From 3 to 7 meals**:
\[
y = 15 - (7 - 3) \times 3 = 15 - 12 = 3 \text{ cups remaining}
\]
- **From 7 to 9 meals**:
\[
y = 3 - (9 - 7) \times 3 = 3 - 6 = -3 \text{ cups (not possible, which indicates an error in meal measurements)}
\]

However, the value should decrease for each subsequent meal, and it appears that only valid measurements are:
- At \( x=1 \), \( y=21 \)
- At \( x=3 \), \( y=15 \)
- At \( x=7 \), \( y=3 \) (as the previous calculation indicated)

Let's summarize how we can derive a linear function based on your findings with valid x-values and y-values. Notably, let's only check the points that progress logically:

- Create a line that connects (1, 21) to (3, 15):

The slope \( m \) between these points:
\[
m = \frac{15 - 21}{3 - 1} = \frac{-6}{2} = -3
\]
Using point-slope form, or simply the linear equation:
\[
y = mx + b
\]

Using point (1, 21) to find b:
\[
21 = -3(1) + b \implies b = 24
\]

The equation at this point is:
\[
y = -3x + 24
\]

We can check this with the other two points only, but assume some alteration in meal sizes with non-linear constraints due to inconsistency, but the modeling could suffice as a linear equation that may require adjustments on consumption logs to adjust for the basis of the last value.

Final concluding function:
\[
y = -3x + 24
\]

With \( y \) as the amount of remaining cups and \( x \) as the number of meals dispensed. Further sampling of values should clarify whether an adjustment for dispensing behavior is necessary.