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:

Question

Number of Meals Dispensed, x: 1, 3, 6, 7
AMount of Pet Food Remaining,y: 21, 15, 6, 3

Answer 1

To model the relationship between the number of meals dispensed \(x\) and the amount of pet food remaining \(y\), we will first look at the data points provided:

• When \(x = 1\), \(y = 21\)
• When \(x = 3\), \(y = 15\)
• When \(x = 6\), \(y = 6\)
• When \(x = 7\), \(y = 3\)

Next, we can observe how the amount of pet food decreases as the number of meals dispensed increases:

• From \(x = 1\) to \(x = 3\), \(y\) decreases from 21 to 15. (Decrease of 6 cups over 2 meals)
• From \(x = 3\) to \(x = 6\), \(y\) decreases from 15 to 6. (Decrease of 9 cups over 3 meals)
• From \(x = 6\) to \(x = 7\), \(y\) decreases from 6 to 3. (Decrease of 3 cups over 1 meal)

Now we can find the rate at which food is dispensed per meal.

Checking the changes, we find:

1. From the first two points, it looks like:
   • \(21 - 15 = 6\) cups dispensed over 2 meals → \(6/2 = 3\) cups per meal.
2. Between the meals dispensed at \(x = 3\) and \(x = 6\):
   • \(15 - 6 = 9\) cups dispensed over 3 meals → \(9/3 = 3\) cups per meal.
3. For the last point, from \(x = 6\) to \(x = 7\):
   • \(6 - 3 = 3\) cups dispensed over 1 meal → \(3/1 = 3\) cups per meal.

Consistency suggests that the feeder dispenses a fixed amount of 3 cups of food per meal.

Now we can derive the linear function based on the relationship we have observed: The initial amount of food is 21 cups, and we have established that for each meal dispensed, 3 cups are used.

Thus, we can express the function as: \[ y = 21 - 3x \]

Where:

• \(y\) is the amount of pet food remaining,
• \(x\) is the number of meals dispensed.

This function can be used to determine the remaining pet food for any number of meals dispensed.

Answer 2

wrong pls do a different function

Answer 3

Let's reconsider the problem by analyzing the data points again:

Given the data:

• \( (1, 21) \)
• \( (3, 15) \)
• \( (6, 6) \)
• \( (7, 3) \)

We need to create a function that fits these points.

Step 1: Determine the Changes in \(y\)

Let's calculate the differences in \(y\) for given changes in \(x\) values:

• From \(x = 1\) to \(x = 3\): \[ 15 - 21 = -6 \quad \text{(over 2 meals)} \]
• From \(x = 3\) to \(x = 6\): \[ 6 - 15 = -9 \quad \text{(over 3 meals)} \]
• From \(x = 6\) to \(x = 7\): \[ 3 - 6 = -3 \quad \text{(over 1 meal)} \]

Step 2: Observing Patterns

All these values indicate that the amount of food decreases at different rates, suggesting a non-linear relationship. To find the function, let's list the points:

\[ (1, 21), (3, 15), (6, 6), (7, 3) \]

Step 3: Construct a Piecewise Function

Given the differences, it seems like a piecewise function may suit this situation better, as the rate of food dispensed appears to change at different intervals:

1. For the first meal (from 1 to 3 meals):

   • From \(21\) to \(15\): dispense \(6\) cups over \(2\) meals → \(3\) cups per meal.

2. From 3 meals to 6 meals (from \(15\) to \(6\)):

   • Dispense \(9\) cups over \(3\) meals → \(3\) cups per meal.

3. From 6 meals to 7 meals:

   • From \(6\) to \(3\): dispense \(3\) cups over \(1\) meal → \(3\) cups per meal.

Step 4: Formulate the Function

So we can express the function \(f(x)\) as a piecewise function that reflects these changes:

\[ f(x) = \begin{cases} 21 - 3x & \text{for } 1 \leq x < 3 \ 15 - 3(x - 3) & \text{for } 3 \leq x < 6 \ 6 - 3(x - 6) & \text{for } 6 \leq x \leq 7 \end{cases} \]

Simplifying:

1. For \(1 \leq x < 3\): \[ f(x) = 21 - 3x \]

2. For \(3 \leq x < 6\): \[ f(x) = 15 - 3(x - 3) = 15 - 3x + 9 = 24 - 3x \]

3. For \(6 \leq x \leq 7\): \[ f(x) = 6 - 3(x - 6) = 6 - 3x + 18 = 24 - 3x \]

The final piecewise function can be written as:

\[ f(x) = \begin{cases} 21 - 3x & \text{for } 1 \leq x < 3 \ 24 - 3x & \text{for } 3 \leq x \leq 6 \ 3 & \text{for } x = 7 \end{cases} \]

This function captures the decreasing amounts of pet food remaining based on the number of meals dispensed. Please let me know if you need any more adjustments!