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Zorluk: OrtaAlgebraic Word Problems and Modeling

A food processing plant packages grain using two automated lines, Line A and Line B. Line A operates at a constant rate of 4040 bags per minute, and Line B operates at a constant rate of 6060 bags per minute. Line A begins filling a batch of 4,2004,200 bags alone. After 1515 minutes, Line A's rate decreases by 25%25\%, and Line B is turned on to work alongside Line A until the batch is complete. How many total minutes does Line A operate to complete the batch?

  1. A
    40 minutes
  2. B
    51 minutes
  3. 55 minutesCevap
  4. D
    63 minutes
  5. E
    70 minutes

Cevap

55 minutes
In the first 15 minutes, Line A fills 40×15=60040 \times 15 = 600 bags, leaving 4,200−600=3,6004,200 - 600 = 3,600 bags remaining. Line A's rate then drops to 40×0.75=3040 \times 0.75 = 30 bags per minute. Working together with Line B (6060 bags per minute), their combined rate is 30+60=9030 + 60 = 90 bags per minute. Filling the remaining 3,6003,600 bags takes 3,60090=40\frac{3,600}{90} = 40 minutes. Thus, Line A operates for a total of 15+40=5515 + 40 = 55 minutes.

Adım Adım Çözüm

1
Calculate the number of bags filled by Line A during the first 15 minutes.
Line A filled 40 bags/min×15 min=600 bags40 \text{ bags/min} \times 15 \text{ min} = 600 \text{ bags}.
Line A worked alone for 15 minutes at its initial constant rate.
2
Determine the remaining number of bags to be filled.
Remaining bags = 4,200−600=3,600 bags4,200 - 600 = 3,600 \text{ bags}.
Subtract the bags already completed from the total batch size.
3
Calculate Line A's reduced rate and the new combined rate of both lines.
Line A's new rate = 40×(1−0.25)=30 bags/min40 \times (1 - 0.25) = 30 \text{ bags/min}. Combined rate = 30+60=90 bags/min30 + 60 = 90 \text{ bags/min}.
Line A's filling rate drops by 25%, and Line B contributes 60 bags per minute simultaneously.
4
Calculate the time needed for both lines to finish the remaining bags.
Time for second phase = 3,600 bags90 bags/min=40 minutes\frac{3,600 \text{ bags}}{90 \text{ bags/min}} = 40 \text{ minutes}.
Divide remaining work by the combined filling rate.
5
Calculate the total time Line A was operating.
Total operating time = 15 min+40 min=55 minutes15 \text{ min} + 40 \text{ min} = 55 \text{ minutes}.
Line A operated during both the initial 15-minute period and the subsequent 40-minute period.

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Algebraic Word Problems and Modeling

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