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 dispensed: 1,3,6,7
amount of pet food remaning: 21,15,6,3
based on the table, which function models this situation?
number of meals dispensed: 1,3,6,7
amount of pet food remaning: 21,15,6,3
based on the table, which function models this situation?
Answers
GPT-4o mini
To find the function that models the situation of pet food in the automatic feeder as a function of the number of meals dispensed, we will first look at the data provided:
- For 1 meal dispensed, there are 21 cups remaining.
- For 3 meals dispensed, there are 15 cups remaining.
- For 6 meals dispensed, there are 6 cups remaining.
- For 7 meals dispensed, there are 3 cups remaining.
Next, we can organize this data in tabular form:
| Meals Dispensed (x) | Food Remaining (y) |
|---------------------|---------------------|
| 1 | 21 |
| 3 | 15 |
| 6 | 6 |
| 7 | 3 |
### Analyzing the Data
To look for a pattern, we can examine how the amount of food changes as meals are dispensed:
1. From 1 to 3 meals:
- Change in meals: \(3 - 1 = 2\)
- Change in food: \(15 - 21 = -6\)
Thus, the average decrease in food per meal is \(-6/2 = -3\).
2. From 3 to 6 meals:
- Change in meals: \(6 - 3 = 3\)
- Change in food: \(6 - 15 = -9\)
Thus, the average decrease in food per meal is \(-9/3 = -3\).
3. From 6 to 7 meals:
- Change in meals: \(7 - 6 = 1\)
- Change in food: \(3 - 6 = -3\)
Thus, the average decrease in food per meal is \(-3/1 = -3\).
From our analysis, it appears that the food decreases consistently by about 3 cups for each meal dispensed.
### Finding the Function
To determine a linear function, we can use the points to identify the slope and intercept. Let's consider the pattern:
We have a decrease of 3 cups per meal dispensed. This will give us the slope \(m = -3\).
Now, we can use the point (1, 21) to find the y-intercept \(b\):
\[ y = mx + b \]
\[ 21 = -3(1) + b \]
\[ 21 = -3 + b \]
\[ b = 21 + 3 = 24 \]
Thus, we can establish the function:
\[ y = -3x + 24 \]
### Conclusion
The function that models the amount of pet food remaining in the automatic feeder based on the number of meals dispensed is:
\[ y = -3x + 24 \]
Where:
- \(y\) is the amount of pet food remaining,
- \(x\) is the number of meals dispensed.
- For 1 meal dispensed, there are 21 cups remaining.
- For 3 meals dispensed, there are 15 cups remaining.
- For 6 meals dispensed, there are 6 cups remaining.
- For 7 meals dispensed, there are 3 cups remaining.
Next, we can organize this data in tabular form:
| Meals Dispensed (x) | Food Remaining (y) |
|---------------------|---------------------|
| 1 | 21 |
| 3 | 15 |
| 6 | 6 |
| 7 | 3 |
### Analyzing the Data
To look for a pattern, we can examine how the amount of food changes as meals are dispensed:
1. From 1 to 3 meals:
- Change in meals: \(3 - 1 = 2\)
- Change in food: \(15 - 21 = -6\)
Thus, the average decrease in food per meal is \(-6/2 = -3\).
2. From 3 to 6 meals:
- Change in meals: \(6 - 3 = 3\)
- Change in food: \(6 - 15 = -9\)
Thus, the average decrease in food per meal is \(-9/3 = -3\).
3. From 6 to 7 meals:
- Change in meals: \(7 - 6 = 1\)
- Change in food: \(3 - 6 = -3\)
Thus, the average decrease in food per meal is \(-3/1 = -3\).
From our analysis, it appears that the food decreases consistently by about 3 cups for each meal dispensed.
### Finding the Function
To determine a linear function, we can use the points to identify the slope and intercept. Let's consider the pattern:
We have a decrease of 3 cups per meal dispensed. This will give us the slope \(m = -3\).
Now, we can use the point (1, 21) to find the y-intercept \(b\):
\[ y = mx + b \]
\[ 21 = -3(1) + b \]
\[ 21 = -3 + b \]
\[ b = 21 + 3 = 24 \]
Thus, we can establish the function:
\[ y = -3x + 24 \]
### Conclusion
The function that models the amount of pet food remaining in the automatic feeder based on the number of meals dispensed is:
\[ y = -3x + 24 \]
Where:
- \(y\) is the amount of pet food remaining,
- \(x\) is the number of meals dispensed.