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Zorluk: ZorAlgebraic Word Problems and Modeling

A manufacturing plant operates two automated production lines, Assembly Line X and Assembly Line Y. Assembly Line Y operates at a standard rate that is 25%25\% faster than the standard rate of Assembly Line X. During a specific shift, Assembly Line X operated for 44 hours at its standard rate, after which its processing rate decreased by 20%20\% for an additional 22 hours due to maintenance. Assembly Line Y began operating 11 hour after Line X started; it operated at its standard rate for 44 hours, and then operated for another 22 hours at 80%80\% of its standard rate. If the two lines produced a combined total of 2,7722,772 units during this shift, what was the standard operating rate of Assembly Line X, in units per hour?

Cevap: 220 units per hour

Cevap

The standard operating rate of Assembly Line X is 220 units per hour.
Assembly Line X produces 4r4r units in its first 4 hours and 1.6r1.6r units in its final 2 hours, totaling 5.6r5.6r units. Assembly Line Y produces 4×1.25r=5r4 \times 1.25r = 5r units in its first 4 hours and 2×(0.80×1.25r)=2r2 \times (0.80 \times 1.25r) = 2r units in its final 2 hours, totaling 7r7r units. The sum of their outputs is 5.6r+7r=12.6r=2,7725.6r + 7r = 12.6r = 2,772. Dividing 2,7722,772 by 12.612.6 gives r=220r = 220.

Adım Adım Çözüm

1
Set up rate expressions for both assembly lines using a single variable
Standard rate of Line X = rr; Standard rate of Line Y = 1.25r1.25r
Line Y is 25% faster than Line X, so its rate is r+0.25r=1.25rr + 0.25r = 1.25r.
2
Calculate the total work done by Assembly Line X
Line X output = 4(r)+2(0.80r)=5.6r4(r) + 2(0.80r) = 5.6r units
Line X worked 4 hours at 100% rate and 2 hours at 80% rate.
3
Calculate the total work done by Assembly Line Y
Line Y output = 4(1.25r)+2(0.80×1.25r)=5r+2r=7r4(1.25r) + 2(0.80 \times 1.25r) = 5r + 2r = 7r units
Line Y worked 4 hours at full rate 1.25r1.25r and 2 hours at 80% of 1.25r1.25r, which equals rate rr.
4
Equate combined production to 2,772 units and solve for rr
5.6r+7r=12.6r=2,772  ⟹  r=2205.6r + 7r = 12.6r = 2,772 \implies r = 220
Dividing the total combined output by the total rate multiplier 12.6 yields the baseline standard rate.

Anahtar Kavram

Algebraic Modeling of Staggered Work and Variable Production Rates
Tahmini Süre:2m 30s
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Algebraic Word Problems and Modeling

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