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.
number of meals dispersed, x | 2, 5, 11
amount of pet food remaining, y | 33, 27, 15

Write a function rule that models this situation.

Write a function rule that models

Answer 1

To create a function that models the relationship between the number of meals dispensed (\(x\)) and the amount of pet food remaining (\(y\)), we can analyze the data points given.

The data points from the table are:

- \((2, 33)\)
- \((5, 27)\)
- \((11, 15)\)

First, let's observe how the amount of pet food decreases as more meals are dispensed.

1. From \(x = 2\) to \(x = 5\):
- Change in meals: \(5 - 2 = 3\)
- Change in food: \(33 - 27 = 6\)

So, for every meal dispensed, the food decreases by:
\[
\text{Rate} = \frac{6}{3} = 2 \text{ cups per meal}
\]

2. From \(x = 5\) to \(x = 11\):
- Change in meals: \(11 - 5 = 6\)
- Change in food: \(27 - 15 = 12\)

So, again the food decreases by:
\[
\text{Rate} = \frac{12}{6} = 2 \text{ cups per meal}
\]

It appears that the rate of decrease is constant, suggesting a linear relationship.

Using the slope-intercept form of a linear equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:

- We found that the slope \(m = -2\) (the food decreases by 2 cups for each meal dispensed).

To find the y-intercept \(b\), we can use one of the points. Let's use the point \( (2, 33) \):

\[
y = mx + b
\]
\[
33 = -2(2) + b
\]
\[
33 = -4 + b
\]
\[
b = 33 + 4 = 37
\]

Now we can write the function rule:

\[
y = -2x + 37
\]

Therefore, the function rule that models this situation is:

\[
\boxed{y = -2x + 37}
\]