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:

y=

Answer 1

To model the situation based on the given data points, we can first analyze the points provided for \(x\) and \(y\):

1. For \(x = 2\), \(y = 33\)
2. For \(x = 5\), \(y = 27\)
3. For \(x = 11\), \(y = 15\)

We can find a linear function of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

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

The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slope between the points (2, 33) and (5, 27):

\[
m = \frac{27 - 33}{5 - 2} = \frac{-6}{3} = -2
\]

Next, let's check the slope between (5, 27) and (11, 15):

\[
m = \frac{15 - 27}{11 - 5} = \frac{-12}{6} = -2
\]

Since the slope is consistent, \(m = -2\).

### Step 2: Determine the y-intercept (b)

Using one of the points to find \(b\), we can use the point (2, 33):

\[
y = mx + b
\]

Substituting \(x = 2\), \(y = 33\), and \(m = -2\):

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

### Step 3: Write the function rule

Now that we have \(m\) and \(b\), the function rule that models this situation is:

\[
y = -2x + 37
\]

### Summary

Thus, the function rule that models the amount of pet food remaining in cups as a function of the number of meals dispensed is:

\[
y = -2x + 37
\]