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Zorluk: ZorSimultaneous Equations and Systems

A telecommunications satellite utilizes two distinct signal amplifier modules, Module A and Module B, to process high-frequency and low-frequency data streams. Each Module A processes 18 Gbps18\text{ Gbps} of high-frequency data and 12 Gbps12\text{ Gbps} of low-frequency data, consuming 150 watts150\text{ watts} of operational power. Each Module B processes 30 Gbps30\text{ Gbps} of high-frequency data and 6 Gbps6\text{ Gbps} of low-frequency data, consuming 180 watts180\text{ watts} of operational power. During a peak transmission test, the active modules processed a combined total of 354 Gbps354\text{ Gbps} of high-frequency data and 138 Gbps138\text{ Gbps} of low-frequency data. What is the total operational power, in watts, consumed by all the active modules during this test?

Cevap: 2460 watts

Cevap

The total operational power consumed by all active modules during the test is 2,460 watts.
By setting up a system of simultaneous equations representing the total high-frequency (18a+30b=35418a + 30b = 354) and low-frequency (12a+6b=13812a + 6b = 138) bandwidths, solving yields a=8a = 8 active Module A units and b=7b = 7 active Module B units. Substituting these module counts into the total power equation 150(8)+180(7)150(8) + 180(7) gives exactly 2,460 watts.

Adım Adım Çözüm

1
Define variables for the unknown counts of modules.
Let aa represent the number of Module A units and bb represent the number of Module B units.
Establishing explicit variables allows representing the data throughput and power constraints as algebraic equations.
2
Formulate equations for high-frequency and low-frequency data streams based on total throughput.
High-frequency equation: 18a+30b=35418a + 30b = 354. Low-frequency equation: 12a+6b=13812a + 6b = 138.
Summing the data contributions from each module type gives the total processed bandwidth for each frequency band.
3
Solve the system of linear equations.
Simplifying the low-frequency equation yields 2a+b=23  ⟹  b=23−2a2a + b = 23 \implies b = 23 - 2a. Substituting into the simplified high-frequency equation (3a+5b=593a + 5b = 59) gives 3a+5(23−2a)=59  ⟹  −7a=−56  ⟹  a=83a + 5(23 - 2a) = 59 \implies -7a = -56 \implies a = 8. Substituting a=8a = 8 back into b=23−2(8)b = 23 - 2(8) gives b=7b = 7.
Determining the exact number of active modules of each type is required before calculating overall energy usage.
4
Calculate the total power consumption.
Total Power = 150a+180b=150(8)+180(7)=1,200+1,260=2,460 watts150a + 180b = 150(8) + 180(7) = 1,200 + 1,260 = 2,460\text{ watts}.
Multiplying the module counts by their respective wattage ratings yields the total operational power.

Anahtar Kavram

Solving systems of simultaneous linear equations with two unknowns and applying the solution to calculate a weighted linear total.
Tahmini Süre:2m 30s
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Simultaneous Equations and Systems

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