A regional logistics node routes cargo containers through a three-stage automated sorting … — GMAT

A regional logistics node routes cargo containers through a three-stage automated sorting process ( $k = 1, 2, 3$ ). At the start of Stage 1 (state $k = 0$ ), the node holds $V_0 = 120$ heavy units and $W_0 = 80$ light units. For each stage $k \in \{1, 2, 3\}$ , the counts of heavy units ( $V_k$ ) and light units ( $W_k$ ) transition according to the following rules:

- $V_k = \frac{1}{2} V_{k-1} + 20$
- $W_k = W_{k-1} + \frac{1}{4} V_{k-1} - 10$

Which of the following statements regarding the state of the cargo units across these three stages are correct?

At the end of Stage 2 ( $k = 2$ ), the number of light units is $110$ .Cevap
The total number of units (heavy plus light) at the end of Stage 3 ( $k = 3$ ) is $165$ .Cevap
C
At the end of Stage 1 ( $k = 1$ ), the number of heavy units exceeds the number of light units by $20$ .
D
The number of heavy units at the end of Stage 3 ( $k = 3$ ) is $60$ .
E
At the end of Stage 2 ( $k = 2$ ), the total number of units is $180$ .

Cevap

The correct statements are that at the end of Stage 2 (

k = 2

), the number of light units is

110

, and the total number of units (heavy plus light) at the end of Stage 3 (

k = 3

) is

165

Evaluating the multi-stage updates step by step shows that at Stage 2 (

k=2

W_2 = 110

, making the statement about 110 light units at Stage 2 correct. At Stage 3 (

k=3

V_3 = 50

and

W_3 = 115

, giving a total of

50 + 115 = 165

units, making the total unit statement at Stage 3 correct.

Adım Adım Çözüm

Calculate state values at the end of Stage 1 (

k = 1

)

V_1 = \frac{1}{2}(120) + 20 = 80

;

W_1 = 80 + \frac{1}{4}(120) - 10 = 100

. Total units

T_1 = 80 + 100 = 180

Apply the given transition rules to initial values

V_0 = 120

and

W_0 = 80

Calculate state values at the end of Stage 2 (

k = 2

)

V_2 = \frac{1}{2}(80) + 20 = 60

;

W_2 = 100 + \frac{1}{4}(80) - 10 = 110

. Total units

T_2 = 60 + 110 = 170

Apply transition rules to

V_1 = 80

and

W_1 = 100

Calculate state values at the end of Stage 3 (

k = 3

)

V_3 = \frac{1}{2}(60) + 20 = 50

;

W_3 = 110 + \frac{1}{4}(60) - 10 = 115

. Total units

T_3 = 50 + 115 = 165

Apply transition rules to

V_2 = 60

and

W_2 = 110

Evaluate each candidate statement against the computed values

Statement regarding light units being 110 at Stage 2 is TRUE (

W_2 = 110

). Statement regarding total units being 165 at Stage 3 is TRUE (

T_3 = 165

). All other statements are false.

Compare calculated stage states directly with the statement claims.

Anahtar Kavram

Sequential state transitions involve iteratively updating state variables through defined algebraic relationships across multiple stages.

Soru

Cevap

Adım Adım Çözüm

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