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 remaining, y
21
15
6
3
Write a function rule that models this situation
Number of Meals Dispensed
1
3
6
7
Amount of Pet food remaining, y
21
15
6
3
Write a function rule that models this situation
Answers
GPT-4o mini
To model the situation described, we can observe the data given:
- Number of Meals Dispensed (x): 1, 3, 6, 7
- Amount of Pet Food Remaining (y in cups): 21, 15, 6, 3
Let's analyze the data points:
- For 1 meal dispensed: 21 cups remaining
- For 3 meals dispensed: 15 cups remaining
- For 6 meals dispensed: 6 cups remaining
- For 7 meals dispensed: 3 cups remaining
We can see that each meal dispensed results in a certain amount of food being used.
To find the change in the amount of pet food remaining as meals are dispensed, let's calculate the change in y (amount of pet food) relative to the change in x (meals dispensed):
- From 1 to 3 meals: Food decreases from 21 to 15 (change of 6 cups)
- From 3 to 6 meals: Food decreases from 15 to 6 (change of 9 cups)
- From 6 to 7 meals: Food decreases from 6 to 3 (change of 3 cups)
This shows that the amount of pet food does not decrease linearly, but we can see that it decreases as more meals are dispensed. We can derive a function that relates the remaining food based on the input.
If we look for a linear trend, we can investigate the pattern of decrease:
Taking two critical points (1, 21) and (7, 3):
- Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 21}{7 - 1} = \frac{-18}{6} = -3 \)
This suggests that for every meal dispensed, the remaining pet food decreases by 3 cups on average.
Now we can write the equation of the line in point-slope form, and we will convert it to slope-intercept form:
1. Using the point (1, 21):
\[
y - 21 = -3(x - 1)
\]
\[
y - 21 = -3x + 3
\]
\[
y = -3x + 24
\]
Thus, the function rule that models the situation is:
\[
y = -3x + 24
\]
Where:
- \( y \) is the amount of pet food remaining,
- \( x \) is the number of meals dispensed.
- Number of Meals Dispensed (x): 1, 3, 6, 7
- Amount of Pet Food Remaining (y in cups): 21, 15, 6, 3
Let's analyze the data points:
- For 1 meal dispensed: 21 cups remaining
- For 3 meals dispensed: 15 cups remaining
- For 6 meals dispensed: 6 cups remaining
- For 7 meals dispensed: 3 cups remaining
We can see that each meal dispensed results in a certain amount of food being used.
To find the change in the amount of pet food remaining as meals are dispensed, let's calculate the change in y (amount of pet food) relative to the change in x (meals dispensed):
- From 1 to 3 meals: Food decreases from 21 to 15 (change of 6 cups)
- From 3 to 6 meals: Food decreases from 15 to 6 (change of 9 cups)
- From 6 to 7 meals: Food decreases from 6 to 3 (change of 3 cups)
This shows that the amount of pet food does not decrease linearly, but we can see that it decreases as more meals are dispensed. We can derive a function that relates the remaining food based on the input.
If we look for a linear trend, we can investigate the pattern of decrease:
Taking two critical points (1, 21) and (7, 3):
- Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 21}{7 - 1} = \frac{-18}{6} = -3 \)
This suggests that for every meal dispensed, the remaining pet food decreases by 3 cups on average.
Now we can write the equation of the line in point-slope form, and we will convert it to slope-intercept form:
1. Using the point (1, 21):
\[
y - 21 = -3(x - 1)
\]
\[
y - 21 = -3x + 3
\]
\[
y = -3x + 24
\]
Thus, the function rule that models the situation is:
\[
y = -3x + 24
\]
Where:
- \( y \) is the amount of pet food remaining,
- \( x \) is the number of meals dispensed.