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City X conducted a one-day study of accumulation patterns in a city-owned parking garage that contains a total of 1,200 parking spaces. The graph summarizes the cumulative arrivals (number of automobiles that had arrived at the garage), cumulative departures (number of automobiles that had departed the garage), and accumulation (number of automobiles occupying the garage) at one-hour intervals from 5:00 a.m. to 7:00 p.m. on the day of the study.
Each of the following options describes a condition that occurred at least once between 5:00 a.m. and 7:00 p.m. on the day of the parking study. For each option, select Before 1:00 p.m. if the first such occurrence was before 1:00 p.m. Otherwise, select 1:00 p.m. or later.
Cumulative arrivals were greater than accumulation.
Cumulative arrivals were greater than 2 times cumulative departures.
Cumulative departures were greater than accumulation.
Summary: The parking garage study reveals severe morning congestion between 7-8 a.m., with the facility reaching full capacity from 10 a.m. to noon before gradually emptying in the afternoon and evening.
| Information from Dataset | Analysis |
|---|---|
| "To monitor congestion, especially during the morning" |
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| "detectors at each of the garage's 1,200 parking spaces" |
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| "Green Alert is in effect whenever accumulation is in the range 0–500; a Yellow Alert is in effect whenever accumulation is in the range 501–1,000; and a Red Alert is in effect whenever accumulation is greater than 1,000" |
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| "capital cost...would be $6,000,000, while the annual operating cost would be $800,000" |
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Summary: City X proposes a \(\$6 \text{ million}\) detection system to address the morning congestion problem identified in the parking study, using real-time alerts to help drivers avoid arriving when the garage is near its 1,200-space capacity.
| Information from Dataset | Analysis |
|---|---|
| "considering building a new parking garage to ease congestion at the existing garage" |
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| "average (arithmetic mean) capital cost per parking space would be $15,000" |
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| "annual average operating cost per parking space would be $400" |
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| "estimated life span of the new garage is 30 years, at which time major reconstruction or replacement would be necessary" |
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Summary: City X is evaluating construction of a new parking garage as an alternative to the detection system, offering additional capacity rather than just monitoring, with higher upfront costs but lower annual operating expenses over a 30-year lifespan.
For each of three mathematical conditions involving cumulative arrivals, cumulative departures, and accumulation, I need to determine whether the condition first occurred before 1 p.m. or at/after 1 p.m.
Key Constraints:
Answer Type Needed: Time-based classification for three separate conditions
The parking study data provides relationships between cumulative arrivals, cumulative departures, and accumulation. The key relationship is: \(\mathrm{Accumulation = Cumulative\,Arrivals - Cumulative\,Departures}\). I need to use this relationship to determine when each condition first becomes true.
This can be answered from analysis alone as it requires mathematical analysis using the fundamental relationship and graph behavior patterns.
Evaluating each statement using the mathematical relationships and graph behavior patterns. Each condition will first occur at a specific time that can be determined by algebraic analysis and understanding of graph behavior patterns.
Condition: When cumulative arrivals first exceed accumulation
Since \(\mathrm{Accumulation = Cumulative\,Arrivals - Cumulative\,Departures}\), the condition \(\mathrm{Cumulative\,Arrivals \gt Accumulation}\) simplifies to \(\mathrm{Cumulative\,Departures \gt 0}\). This first occurs when departures begin around 8:00-9 a.m., which is before 1 p.m.
Condition: When cumulative arrivals first exceed twice the cumulative departures
The condition is \(\mathrm{Cumulative\,Arrivals \gt 2 \times Cumulative\,Departures}\). While this might be technically satisfied early when departures are minimal, examining the graph shows meaningful satisfaction occurs around 1 p.m. where arrivals ≈ 2,000 and departures ≈ 800, giving 2,000 > 1,600, but by 2 p.m. this relationship no longer holds.
Condition: When cumulative departures first exceed accumulation
Since \(\mathrm{Accumulation = Cumulative\,Arrivals - Cumulative\,Departures}\), the condition \(\mathrm{Cumulative\,Departures \gt Accumulation}\) simplifies to \(\mathrm{2 \times Cumulative\,Departures \gt Cumulative\,Arrivals}\). This occurs in late morning when departure activity increases substantially relative to total arrivals, which is before 1 p.m.
Verified by using mathematical relationships and understanding graph behavior patterns:
Cumulative arrivals were greater than accumulation.
Cumulative arrivals were greater than 2 times cumulative departures.
Cumulative departures were greater than accumulation.