An industrial semiconductor fabrication facility operates three specialized chemical vapor… — GMAT

An industrial semiconductor fabrication facility operates three specialized chemical vapor deposition (CVD) chambers—Chamber 1, Chamber 2, and Chamber 3—to process a large batch of silicon wafers.

When Chamber 1 and Chamber 2 operate together at their respective constant rates, they can complete the entire batch in $12$ hours. When Chamber 2 and Chamber 3 operate together at their respective constant rates, they can complete the entire batch in $20$ hours.

To process a new batch, Chamber 1 operates alone for $4$ hours. Next, Chamber 2 is turned on to assist Chamber 1, and both chambers operate together for an additional $6$ hours. Finally, Chambers 1 and 2 are shut down, and Chamber 3 operates alone for $18$ hours to finish the remaining portion of the batch.

How many hours would it take Chamber 3 to process the entire batch working alone at its constant rate?

A
40 hours
B
48 hours
C
54 hours
60 hoursCevap
E
72 hours

Cevap

60 hours

The correct answer is 60 hours. By subtracting the combined rate of Chambers 2 and 3 (

\frac{1}{20}

) from the combined rate of Chambers 1 and 2 (

\frac{1}{12}

), we find that Chamber 1's rate exceeds Chamber 3's rate by

\frac{1}{30}

batch per hour. Substituting

r_1 = r_3 + \frac{1}{30}

into the total work equation for the three stages (

4 r_1 + 6(r_1 + r_2) + 18 r_3 = 1

) yields

22 r_3 + \frac{2}{15} = \frac{1}{2}

, giving

r_3 = \frac{1}{60}

. Therefore, Chamber 3 requires 60 hours alone.

Adım Adım Çözüm

Define rate variables and establish equations for the combined pairs.

Let

r_1, r_2, r_3

represent the work rates of Chambers 1, 2, and 3 in fraction of batch per hour. We are given

r_1 + r_2 = \frac{1}{12}

and

r_2 + r_3 = \frac{1}{20}

Work rate is the reciprocal of completion time for a full job.

Find the difference between the rates of Chamber 1 and Chamber 3.

(r_1 + r_2) - (r_2 + r_3) = \frac{1}{12} - \frac{1}{20} \implies r_1 - r_3 = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30}

. Thus,

r_1 = r_3 + \frac{1}{30}

Eliminating

r_2

allows us to express

r_1

strictly in terms of

r_3

Calculate the work completed in each stage of the multi-stage schedule.

Stage 1 work =

4 r_1

. Stage 2 work =

6(r_1 + r_2) = 6 \left(\frac{1}{12}\right) = \frac{1}{2}

. Stage 3 work =

18 r_3

Work completed equals rate multiplied by time operating.

Set up the total work equation and solve for Chamber 3's rate

r_3

4 r_1 + \frac{1}{2} + 18 r_3 = 1 \implies 4 \left(r_3 + \frac{1}{30}\right) + 18 r_3 = \frac{1}{2} \implies 22 r_3 + \frac{2}{15} = \frac{1}{2} \implies 22 r_3 = \frac{15 - 4}{30} = \frac{11}{30} \implies r_3 = \frac{11}{30 \times 22} = \frac{1}{60}

Sum of work across all stages equals

1

complete batch.

Determine the time required for Chamber 3 to complete the entire batch alone.

Time

= \frac{1}{r_3} = 60

hours.

Total time for a single entity is the reciprocal of its individual work rate.

Anahtar Kavram

Multi-stage work rate modeling with system of linear equations

Tahmini Süre:2m 30s

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