A chemical synthesis apparatus monitors two reactant volumes, A_k and B_k (in liters), acr… — GMAT

A chemical synthesis apparatus monitors two reactant volumes, $A_k$ and $B_k$ (in liters), across three sequential stages ( $k = 1, 2, 3$ ). At the initial state ( $k = 0$ ), the apparatus contains $A_0 = 100$ liters and $B_0 = 40$ liters.

For each subsequent stage

k

, the volumes are updated sequentially according to the following state transition equations:

A_k = A_{k-1} - B_{k-1} + 40

B_k = B_{k-1} + \frac{A_{k-1}}{2} - 20

Based on the process above, which of the following statements regarding the reactant volumes are correct? Select all that apply.

At the end of Stage 2 ( $k = 2$ ), the volume of reactant A ( $A_2$ ) is 70 liters.Answer
At the end of Stage 3 ( $k = 3$ ), the volume of reactant B ( $B_3$ ) is 115 liters.Answer
C
At the end of Stage 2 ( $k = 2$ ), the volume of reactant B ( $B_2$ ) is 70 liters.
D
At the end of Stage 3 ( $k = 3$ ), the volume of reactant A ( $A_3$ ) is 70 liters.
E
At the end of Stage 1 ( $k = 1$ ), the volume of reactant B ( $B_1$ ) is 100 liters.

Answer

The correct statements are that at the end of Stage 2, the volume of reactant A is 70 liters, and at the end of Stage 3, the volume of reactant B is 115 liters.

Evaluating the process iteratively stage by stage shows that at

k=2

A_2 = 70

liters, and at

k=3

B_3 = 115

liters. Both of these statements match the calculated values.

Step-by-Step Solution

Calculate state values at the end of Stage 1 (

k=1

)

A_1 = 100 - 40 + 40 = 100

liters;

B_1 = 40 + \frac{100}{2} - 20 = 70

liters.

Apply state equations using initial inputs

A_0 = 100

and

B_0 = 40

Calculate state values at the end of Stage 2 (

k=2

)

A_2 = 100 - 70 + 40 = 70

liters;

B_2 = 70 + \frac{100}{2} - 20 = 100

liters.

Apply state equations using inputs

A_1 = 100

and

B_1 = 70

Calculate state values at the end of Stage 3 (

k=3

)

A_3 = 70 - 100 + 40 = 10

liters;

B_3 = 100 + \frac{70}{2} - 20 = 115

liters.

Apply state equations using inputs

A_2 = 70

and

B_2 = 100

Evaluate the proposed options against calculated values

The statement specifying

A_2 = 70

liters is correct. The statement specifying

B_3 = 115

liters is correct.

Match calculated state outputs with the offered statements.

Key Concept

Sequential Process Execution and Coupled State Transitions

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