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Zorluk: ZorRatios, Rates, and Percentages

A cloud computing facility operates four server clusters—Alpha, Beta, Gamma, and Delta—each processing data tasks at a constant rate while consuming electrical energy. Following a infrastructure optimization, each cluster undergoes modifications to its task processing rate and energy consumption:

- Cluster Alpha: Originally processes 12,000 tasks/hr12,000\text{ tasks/hr} using 60 kWh60\text{ kWh} of energy. After optimization, its task processing rate increases by 25%25\% while its energy consumption decreases by 20%20\%.
- Cluster Beta: Originally processes 16,000 tasks/hr16,000\text{ tasks/hr} using 80 kWh80\text{ kWh} of energy. After optimization, its task processing rate increases by 15%15\% while its energy consumption decreases by 8%8\%.
- Cluster Gamma: Originally processes 20,000 tasks/hr20,000\text{ tasks/hr} using 100 kWh100\text{ kWh} of energy. After optimization, its task processing rate increases by 8%8\% while its energy consumption decreases by 10%10\%.
- Cluster Delta: Originally processes 25,000 tasks/hr25,000\text{ tasks/hr} using 125 kWh125\text{ kWh} of energy. After optimization, its task processing rate increases by 12%12\% while its energy consumption decreases by 20%20\%.

Match each server cluster to its corresponding post-optimization efficiency metrics (new task processing rate and energy efficiency ratio in tasks per kWh).

  • Cluster AlphaNew processing rate of 15,000 tasks/hr15,000\text{ tasks/hr} with an efficiency ratio of 312.5 tasks/kWh312.5\text{ tasks/kWh}
  • Cluster BetaNew processing rate of 18,400 tasks/hr18,400\text{ tasks/hr} with an efficiency ratio of 250 tasks/kWh250\text{ tasks/kWh}
  • Cluster GammaNew processing rate of 21,600 tasks/hr21,600\text{ tasks/hr} with an efficiency ratio of 240 tasks/kWh240\text{ tasks/kWh}
  • Cluster DeltaNew processing rate of 28,000 tasks/hr28,000\text{ tasks/hr} with an efficiency ratio of 280 tasks/kWh280\text{ tasks/kWh}

Cevap

Cluster Alpha matches with 15,000 tasks/hr15,000\text{ tasks/hr} and 312.5 tasks/kWh312.5\text{ tasks/kWh}; Cluster Beta matches with 18,400 tasks/hr18,400\text{ tasks/hr} and 250 tasks/kWh250\text{ tasks/kWh}; Cluster Gamma matches with 21,600 tasks/hr21,600\text{ tasks/hr} and 240 tasks/kWh240\text{ tasks/kWh}; Cluster Delta matches with 28,000 tasks/hr28,000\text{ tasks/hr} and 280 tasks/kWh280\text{ tasks/kWh}.
Each cluster's new task processing rate and new power consumption must be computed independently using their respective growth and reduction multipliers. Dividing the new rate by the new power gives the exact post-optimization efficiency ratio in tasks per kWh for each cluster.

Adım Adım Çözüm

1
Calculate the updated task processing rate for each cluster by applying the respective percentage increase factor.
Rate_Alpha = 12,000×(1+0.25)=15,000 tasks/hr12,000 \times (1 + 0.25) = 15,000\text{ tasks/hr}; Rate_Beta = 16,000×(1+0.15)=18,400 tasks/hr16,000 \times (1 + 0.15) = 18,400\text{ tasks/hr}; Rate_Gamma = 20,000×(1+0.08)=21,600 tasks/hr20,000 \times (1 + 0.08) = 21,600\text{ tasks/hr}; Rate_Delta = 25,000×(1+0.12)=28,000 tasks/hr25,000 \times (1 + 0.12) = 28,000\text{ tasks/hr}.
An increase of p%p\% modifies the original quantity by a factor of (1+p100)(1 + \frac{p}{100}).
2
Calculate the updated energy consumption (kWh) for each cluster by applying the respective percentage reduction factor.
Power_Alpha = 60×(10.20)=48 kWh60 \times (1 - 0.20) = 48\text{ kWh}; Power_Beta = 80×(10.08)=73.6 kWh80 \times (1 - 0.08) = 73.6\text{ kWh}; Power_Gamma = 100×(10.10)=90 kWh100 \times (1 - 0.10) = 90\text{ kWh}; Power_Delta = 125×(10.20)=100 kWh125 \times (1 - 0.20) = 100\text{ kWh}.
A decrease of d%d\% modifies the original energy usage by a factor of (1d100)(1 - \frac{d}{100}).
3
Compute the new power efficiency ratio (tasks per kWh) for each cluster by dividing its new task processing rate by its new energy consumption.
Eff_Alpha = 15,00048=312.5 tasks/kWh\frac{15,000}{48} = 312.5\text{ tasks/kWh}; Eff_Beta = 18,40073.6=250 tasks/kWh\frac{18,400}{73.6} = 250\text{ tasks/kWh}; Eff_Gamma = 21,60090=240 tasks/kWh\frac{21,600}{90} = 240\text{ tasks/kWh}; Eff_Delta = 28,000100=280 tasks/kWh\frac{28,000}{100} = 280\text{ tasks/kWh}.
Efficiency is defined as output rate per unit of energy consumed.
4
Pair each server cluster to its corresponding output description.
Cluster Alpha pairs with 15,000 tasks/hr15,000\text{ tasks/hr} (312.5 tasks/kWh312.5\text{ tasks/kWh}); Cluster Beta pairs with 18,400 tasks/hr18,400\text{ tasks/hr} (250 tasks/kWh250\text{ tasks/kWh}); Cluster Gamma pairs with 21,600 tasks/hr21,600\text{ tasks/hr} (240 tasks/kWh240\text{ tasks/kWh}); Cluster Delta pairs with 28,000 tasks/hr28,000\text{ tasks/hr} (280 tasks/kWh280\text{ tasks/kWh}).
Matching calculated values directly to the corresponding right-side item options.

Anahtar Kavram

Ratios, Rates, and Multi-step Percentage Modifications
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