To analyze the parking garage rates provided in the table:
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Rate of Change: The rate of change can be calculated as the change in cost divided by the change in hours. We can take pairs of points from the provided data:
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From (1 hour, $10) to (3 hours, $14): \[ \text{Change in cost} = 14 - 10 = 4 \] \[ \text{Change in hours} = 3 - 1 = 2 \] \[ \text{Rate of change} = \frac{4}{2} = 2 \]
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From (3 hours, $14) to (5 hours, $16): \[ \text{Change in cost} = 16 - 14 = 2 \] \[ \text{Change in hours} = 5 - 3 = 2 \] \[ \text{Rate of change} = \frac{2}{2} = 1 \]
Since the rate of change differs between these intervals, the overall rate of change is not constant.
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Y-Intercept: The y-intercept is the cost when the number of hours is \(0\). Since there isn't an explicit value given for \(0\) hours in the table and we cannot extrapolate directly from the table data, we may infer a y-intercept based on existing data. When observing the pattern, it would be logical to assume a charge of $10 for first hour and maybe no charges for zero hours.
If we consider a linear equation from the first two points: \[ \text{Equation in slope-intercept form is} \ \text{y} = mx + b \] Using the first point (1, 10) with \(m = 2\) leads us: \[ 10 = 2(1) + b \Rightarrow b = 8 \] So, we could infer the y-intercept might be \( $8\), although based on the context that can also be \(0\).
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Is this relationship proportional? A relationship is proportional if the ratio between the dependent variable and independent variable is constant and passes through the origin. Given that the rate changes (as seen in rate of change calculations), this relationship does not maintain a constant ratio. Additionally, if you take the cost at hours 1, 3 and 5, cost for 0 hours is not proportionate. So, it is not a proportional relationship.
In conclusion:
- Rate of Change: Not constant; varies between intervals.
- Y-Intercept: Possibly $8 (assuming a linear estimation based on the data).
- Proportional Relationship: No, it is not proportional.
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