Two-Part Analysis

Arrange the operating intervals in ascending order based on the battery energy reserve $E_k$ measured at the end of each interval, from the lowest energy reserve to the highest energy reserve.

A pharmaceutical laboratory synthesizes two custom therapeutic compounds, Compound $X$ and Compound $Y$, in integer batch quantities during a single production cycle. Production is subject to the following resource and operational constraints:

- Bioreactor Time: Each batch of Compound $X$ requires $4$ hours and each batch of Compound $Y$ requires $3$ hours. The laboratory has at most $48$ total hours of bioreactor time available.
- Specialized Solvent: Each batch of Compound $X$ requires $3$ liters and each batch of Compound $Y$ requires $5$ liters. The total solvent supply is capped at $55$ liters.
- Catalyst Stability Limit: To prevent reactive degradation, the number of batches of Compound $Y$ produced cannot exceed twice the number of batches of Compound $X$ plus $1$ ($Y \le 2X + 1$).

Each batch of Compound $X$ generates a net profit of $\$12,000$, and each batch of Compound $Y$ generates a net profit of $\$14,000$.

What is the maximum total net profit, in dollars, that the laboratory can achieve within these combined production constraints?

An earth-observation satellite payload operator allocates integer numbers of channels to two operational payload modes: High-Resolution Imaging ($I$) and Atmospheric Sounding ($S$). Operational parameters and resource bounds are specified as follows:

- Bandwidth: Each imaging channel requires $15\text{ MHz}$ and each sounding channel requires $10\text{ MHz}$. Total available payload bandwidth is at most $210\text{ MHz}$.
- Power: Each imaging channel consumes $40\text{ W}$ and each sounding channel consumes $15\text{ W}$. Total available power budget is at most $440\text{ W}$.
- Mission Minimums: The satellite must operate at least $2$ imaging channels ($I \ge 2$) and at least $4$ sounding channels ($S \ge 4$).
- Buffer Constraint: The number of sounding channels cannot exceed twice the number of imaging channels ($S \le 2I$).

Match each payload optimization target on the left with its corresponding integer channel count on the right.

An industrial water desalination facility routes seawater through a three-stage membrane filtration process ($k = 1, 2, 3$). At stage $k = 0$ (the initial feed stage), the water has a salt concentration of $C_0 = 800\text{ ppm}$ and a flow rate of $F_0 = 120\text{ liters/min}$.

At each subsequent stage $k$ (where $k = 1, 2, 3$), the state of the water is updated according to the following sequential rules:
- Salt concentration: $C_k = 0.5 C_{k-1} - 20\text{ ppm}$
- Flow rate: $F_k = F_{k-1} - 15\text{ liters/min}$

Which of the following statements regarding the system's state transitions across the stages are true? Select all that apply.

A telecommunications company deploys two installation teams, Team Alpha and Team Beta, to lay fiber-optic cable across a rural region containing both rocky terrain and flat terrain.

- Team Alpha lays cable at a constant rate of $400$ meters per day in rocky terrain and $600$ meters per day in flat terrain. Over a project lasting exactly $20$ working days, Team Alpha laid a total of $10,400$ meters of cable.
- Team Beta lays cable at a constant rate of $300$ meters per day in rocky terrain and $700$ meters per day in flat terrain. Over a project also lasting exactly $20$ working days, Team Beta laid a total of $11,600$ meters of cable.

Based on the information provided, which of the following correctly pairs the number of days Team Alpha worked in rocky terrain with the number of days Team Beta worked in rocky terrain?

An automated data processing node manages a queue of records across three sequential processing stages ($k = 1, 2, 3$). The initial batch size at stage $k = 0$ is $N_0 = 400$ records. In each stage $k$, the batch size undergoes two consecutive transitions:
1. The batch size is reduced by $50\%$ due to quality filtering.
2. Exactly $120$ newly verified records are added to the batch.

Which of the following statements regarding the batch size across these stages are correct? Select all that apply.

A digital payment platform processes a batch of $N_0 = 1{}000$ transactions through four sequential verification stages ($k = 1, 2, 3, 4$). The transition rules for the transaction count $N_k$ at each stage $k$ are defined as follows:

* Stage 1 (Fraud Filter): $10\%$ of the current batch transactions are flagged and removed, after which $50$ system control transactions are added.
* Stage 2 (Currency Verification): $20\%$ of the current batch transactions are each split into two separate micro-transactions (increasing the net transaction count by $20\%$ of the incoming batch), after which $40$ invalid transactions are purged.
* Stage 3 (Settlement Reconciliation): The batch transaction count decreases by $15\%$ due to netting, after which $150$ pending transactions are merged into the batch.
* Stage 4 (Archival Audit): $20\%$ of the current batch transactions are archived (retaining $80\%$ in the active batch), after which $120$ final clearance tokens are added to the batch.

Place the four stages in order of their final batch transaction counts ($N_k$), from lowest transaction count to highest transaction count.

An automated smart climate controller regulates room temperature ($T$, in ${}^\circ\text{C}$) and relative humidity ($H$, in $\%$) across three sequential hourly cycles ($k = 1, 2, 3$). The initial state at cycle $k = 0$ is $T_0 = 24^\circ\text{C}$ and $H_0 = 60\%$.

At the end of each hourly cycle $k$, the new state $(T_k, H_k)$ is updated according to the following transition rules:
- $T_k = T_{k-1} - 2$ if $H_{k-1} \ge 50\%$; otherwise, $T_k = T_{k-1} + 1$
- $H_k = H_{k-1} - 10$ if $T_{k-1} > 20$; otherwise, $H_k = H_{k-1} + 5$

Which of the following statements regarding the system's state are correct? Select all that apply.

A digital streaming platform processes high-definition video files using two server clusters, Cluster A and Cluster B.

- Cluster A processes video data at a constant rate of $100\text{ GB/hour}$, and $18\%$ of its processed data requires secondary re-encoding.
- Cluster B processes video data at a constant rate of $200\text{ GB/hour}$, and $8\%$ of its processed data requires secondary re-encoding.

During a processing cycle, the platform processed a total of $3,000\text{ GB}$ of video data across both clusters, and exactly $12\%$ of the total processed data required secondary re-encoding.

Which of the following correctly pairs the operating time of Cluster A and the operating time of Cluster B during this cycle?

A municipal water treatment facility operates four filtration basins (Basin A, Basin B, Basin C, and Basin D). Each basin receives raw water at a specified intake rate and operates at a given recovery percentage (the percent of raw water converted to purified water). Match each filtration basin on the left to its corresponding purified water output rate in thousands of gallons per hour (kgal/hr) on the right.

A manufacturing facility operates two assembly lines, Line 1 and Line 2, producing high-precision components.

- Line 1 operates at a constant rate of $R_1$ components per hour, and Line 2 operates at a constant rate of $R_2$ components per hour.
- Operating Line 1 for $5$ hours and Line 2 for $6$ hours yields a total output of $1,200$ components.
- If Line 1's production rate is increased by $25\%$ and Line 2's production rate is decreased by $15\%$, running both lines together for $4$ hours yields a total output of $940$ components.

Which of the following represents the original production rate of Line 1 ($R_1$) and Line 2 ($R_2$), respectively, in components per hour?