To determine the rate of change for the water level over time, we'll examine the differences in water levels between consecutive time points and divide those differences by the time intervals. This will give us the rate of change of water level per unit time.
Here's the step-by-step process:
1. Identify the given data points:
- Time (minutes): [tex]\( [7, 8, 9, 10] \)[/tex]
- Water (gallons): [tex]\( [38.5, 36.25, 34, 31.75] \)[/tex]
2. Calculate the differences in water levels between consecutive time points (ΔWater):
- From time 7 to 8: [tex]\( 36.25 - 38.5 = -2.25 \)[/tex]
- From time 8 to 9: [tex]\( 34 - 36.25 = -2.25 \)[/tex]
- From time 9 to 10: [tex]\( 31.75 - 34 = -2.25 \)[/tex]
3. Calculate the time intervals (ΔTime):
- From time 7 to 8: [tex]\( 8 - 7 = 1 \)[/tex]
- From time 8 to 9: [tex]\( 9 - 8 = 1 \)[/tex]
- From time 9 to 10: [tex]\( 10 - 9 = 1 \)[/tex]
4. Determine the rate of change for each interval by dividing the change in water level by the corresponding time interval:
- Rate of change from 7 to 8: [tex]\( \frac{-2.25}{1} = -2.25 \)[/tex]
- Rate of change from 8 to 9: [tex]\( \frac{-2.25}{1} = -2.25 \)[/tex]
- Rate of change from 9 to 10: [tex]\( \frac{-2.25}{1} = -2.25 \)[/tex]
5. Summarize the results:
The rate of change for each time interval is consistently [tex]\( -2.25 \)[/tex] gallons per minute. This means that the water level is decreasing at a constant rate of [tex]\( 2.25 \)[/tex] gallons per minute as the bathtub is being drained.
Therefore, the rate of change for the relationship between time and water level in the bathtub is [tex]\( \boxed{-2.25 \)[/tex] gallons per minute}.
