An industrial water desalination facility routes seawater through a three-stage membrane f… — GMAT

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.

At the end of Stage 2 ( $k = 2$ ), the salt concentration of the water is $170\text{ ppm}$ .Answer
At the end of Stage 3 ( $k = 3$ ), the water flow rate is $75\text{ liters/min}$ .Answer
C
At the end of Stage 2 ( $k = 2$ ), the salt concentration of the water is $380\text{ ppm}$ .
D
At the end of Stage 3 ( $k = 3$ ), the salt concentration of the water is $170\text{ ppm}$ .
E
At the end of Stage 1 ( $k = 1$ ), the water flow rate is $90\text{ liters/min}$ .

Answer

The correct statements are the one indicating a salt concentration of 170 ppm at Stage 2 and the one indicating a flow rate of 75 liters/min at Stage 3.

Evaluating each transition step sequentially gives C_1 = 380, C_2 = 170, and C_3 = 65 for salt concentration, and F_1 = 105, F_2 = 90, and F_3 = 75 for flow rate. The statement asserting a salt concentration of 170 ppm at Stage 2 and the statement asserting a flow rate of 75 liters/min at Stage 3 correctly reflect these evaluations.

Step-by-Step Solution

Calculate state metrics for Stage 1 (k = 1)

C_1 = 0.5(800) - 20 = 380 ppm; F_1 = 120 - 15 = 105 liters/min

Apply the recursive transition functions once from initial state k = 0.

Calculate state metrics for Stage 2 (k = 2)

C_2 = 0.5(380) - 20 = 170 ppm; F_2 = 105 - 15 = 90 liters/min

Apply the recursive transition functions to the Stage 1 outputs.

Calculate state metrics for Stage 3 (k = 3)

C_3 = 0.5(170) - 20 = 65 ppm; F_3 = 90 - 15 = 75 liters/min

Apply the recursive transition functions to the Stage 2 outputs.

Verify each statement against calculated values

Stage 2 salt concentration is 170 ppm (True) and Stage 3 flow rate is 75 liters/min (True).

Identify all statements that match the calculated sequence values.

Key Concept

Multi-stage recursive state evaluation and linear parameter transition tracking.

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