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Difficulty: MediumAlgebraic Word Problems and Modeling

A municipal utility company charges residential customers according to two seasonal rate structures. During the peak summer season, electricity costs $0.24\$0.24 per kilowatt-hour (kWh\text{kWh}). During the off-peak season, electricity costs $0.15\$0.15 per kWh\text{kWh}. A homeowner consumed a total of 14,000 kWh14,000\text{ kWh} of electricity over the entire year, resulting in a total annual bill of $2,640\$2,640. How many kilowatt-hours of electricity did the homeowner consume during the peak summer season?

  1. 6,0006,000Answer
  2. B
    8,0008,000
  3. C
    2,2502,250
  4. D
    11,00011,000
  5. E
    7,0007,000

Answer

6,000 kWh6,000\text{ kWh}
The correct answer of 6,000 kWh6,000\text{ kWh} is found by setting up a linear model for total energy consumption (x+y=14,000x + y = 14,000) and total cost (0.24x+0.15y=2,6400.24x + 0.15y = 2,640). Substituting y=14,000−xy = 14,000 - x gives 0.09x=5400.09x = 540, which solves to x=6,000 kWhx = 6,000\text{ kWh}.

Step-by-Step Solution

1
Define variables for the two seasons.
Let xx be the number of kilowatt-hours consumed during the peak summer season, and let yy be the number of kilowatt-hours consumed during the off-peak season.
Establishing clear variables allows modeling the problem as a system of linear equations.
2
Set up the total consumption equation and express yy in terms of xx.
x+y=14,000  ⟹  y=14,000−xx + y = 14,000 \implies y = 14,000 - x
The total consumption across both seasons equals 14,000 kWh14,000\text{ kWh}.
3
Set up the total cost equation using the respective seasonal rates.
0.24x+0.15y=2,6400.24x + 0.15y = 2,640
Peak electricity costs $0.24/kWh\$0.24/\text{kWh} and off-peak costs $0.15/kWh\$0.15/\text{kWh}, summing to the total annual cost of $2,640\$2,640.
4
Substitute y=14,000−xy = 14,000 - x into the cost equation and solve for xx.
0.24x+0.15(14,000−x)=2,640  ⟹  0.24x+2,100−0.15x=2,640  ⟹  0.09x=540  ⟹  x=6,0000.24x + 0.15(14,000 - x) = 2,640 \implies 0.24x + 2,100 - 0.15x = 2,640 \implies 0.09x = 540 \implies x = 6,000
Simplifying the single-variable linear equation isolates the peak season consumption.

Key Concept

Linear Modeling and Systems of Linear Equations
Estimated Time:1m 30s
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Topics

Algebraic Word Problems and Modeling

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