To determine the rate of change for the function given in the table, we need to find the change in the number of grapes eaten in relation to the change in time. This rate of change can be interpreted as the slope of the line that represents the relationship between time and the number of grapes eaten.
The formula for the rate of change (slope) is:
[tex]\[ \text{Rate of Change} = \frac{\Delta y}{\Delta x} \][/tex]
Where:
- [tex]\( \Delta y \)[/tex] is the change in the number of grapes eaten.
- [tex]\( \Delta x \)[/tex] is the change in time.
Let's choose two points from the table to calculate the rate of change. For instance, we can use the points (1, 15) and (2, 30).
1. Identify the coordinates of the two points:
- First point: [tex]\( (x_1, y_1) = (1, 15) \)[/tex]
- Second point: [tex]\( (x_2, y_2) = (2, 30) \)[/tex]
2. Calculate the change in the number of grapes eaten ([tex]\( \Delta y \)[/tex]):
[tex]\[ \Delta y = y_2 - y_1 = 30 - 15 = 15 \][/tex]
3. Calculate the change in time ([tex]\( \Delta x \)[/tex]):
[tex]\[ \Delta x = x_2 - x_1 = 2 - 1 = 1 \][/tex]
4. Substitute these values into the rate of change formula:
[tex]\[ \text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{15}{1} = 15 \][/tex]
So, the rate of change is 15 grapes eaten per minute.
Among the given options:
- 15 grapes eaten per minute
- 15 minutes to eat each grape
- 60 grapes eaten per minute
- 60 minutes to eat each grape
The correct answer is:
15 grapes eaten per minute.
