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Zorluk: OrtaAlgebraic Word Problems and Modeling

Two industrial processing units, Unit X and Unit Y, process liquid solution at constant rates of 4040 liters per hour and 6060 liters per hour, respectively. Unit X begins processing a 1,1001,100-liter batch of solution alone. After 55 hours, Unit Y is activated and joins Unit X, working simultaneously at their respective constant rates until the entire batch is completely processed. What is the total time, in hours, from when Unit X started processing until the entire batch was completely processed?

  1. A
    9
  2. B
    11
  3. 14Cevap
  4. D
    16
  5. E
    20

Cevap

14 hours
Unit X works alone for 5 hours, processing 40×5=20040 \times 5 = 200 liters. This leaves 1,100−200=9001,100 - 200 = 900 liters of solution remaining. When Unit Y joins, the units process solution at a joint rate of 40+60=10040 + 60 = 100 liters per hour. The remaining 900 liters require 900100=9\frac{900}{100} = 9 hours of combined work. Summing the 5 hours of solo work and 9 hours of combined work gives a total time of 14 hours.

Adım Adım Çözüm

1
Calculate the volume processed by Unit X during its solo operation
40 liters/hour×5 hours=200 liters40 \text{ liters/hour} \times 5 \text{ hours} = 200 \text{ liters}
Unit X operates alone for the first 5 hours at a constant rate of 40 liters per hour.
2
Determine the remaining batch volume to be processed
1,100 liters−200 liters=900 liters1,100 \text{ liters} - 200 \text{ liters} = 900 \text{ liters}
Subtract the volume already completed from the total batch size.
3
Calculate the combined processing rate of Unit X and Unit Y
40 liters/hour+60 liters/hour=100 liters/hour40 \text{ liters/hour} + 60 \text{ liters/hour} = 100 \text{ liters/hour}
Both units work together simultaneously after the first 5 hours.
4
Find the time needed for both units to process the remaining volume
900 liters100 liters/hour=9 hours\frac{900 \text{ liters}}{100 \text{ liters/hour}} = 9 \text{ hours}
Divide the remaining volume by the combined rate.
5
Calculate the total elapsed time from the start
5 hours (solo)+9 hours (combined)=14 hours5 \text{ hours (solo)} + 9 \text{ hours (combined)} = 14 \text{ hours}
Add the initial solo work time to the combined work time.

Anahtar Kavram

Algebraic modeling of combined work rates with staggered start times
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Algebraic Word Problems and Modeling

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