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Difficulty: HardBasic Algebraic Expressions and One-Step Equations

A solar power microgrid generates electricity at a constant rate of pp kilowatts for each hour of active operation. On a day with hh total hours of daylight, cloud coverage reduces operation such that the microgrid operates at full capacity for 33 hours less than half the total daylight hours. Which of the following equations represents the total electricity EE, in kilowatt-hours, generated by the microgrid on that day?

  1. E=p(12h−3)E = p\left(\frac{1}{2}h - 3\right)Answer
  2. B
    E=p(3−12h)E = p\left(3 - \frac{1}{2}h\right)
  3. C
    E=12ph−3E = \frac{1}{2}ph - 3
  4. D
    E=p(h−32)E = p\left(\frac{h - 3}{2}\right)
  5. E
    E=p(12h+3)E = p\left(\frac{1}{2}h + 3\right)

Answer

The correct equation is E=p(12h−3)E = p\left(\frac{1}{2}h - 3\right).
The expression for half the total daylight hours is 12h\frac{1}{2}h. Taking '3 hours less than' this quantity requires subtracting 3, yielding an active time of (12h−3)\left(\frac{1}{2}h - 3\right) hours. Multiplying the hourly power rate pp by this active time gives the total energy equation E=p(12h−3)E = p\left(\frac{1}{2}h - 3\right).

Step-by-Step Solution

1
Express half of the total daylight hours algebraically.
Half of hh total hours is written as 12h\frac{1}{2}h or h2\frac{h}{2}.
Dividing the total daylight duration hh by 2 establishes the baseline time frame.
2
Translate '3 hours less than half the total daylight hours' into a time expression.
Operating time =12h−3= \frac{1}{2}h - 3.
The phrase '3 less than X' means X−3X - 3, so 3 is subtracted from 12h\frac{1}{2}h.
3
Multiply the rate of generation by the active operating time to form the equation for total electricity EE.
E=p(12h−3)E = p\left(\frac{1}{2}h - 3\right).
Total energy generated is equal to the hourly rate pp multiplied by the total active operating hours.

Key Concept

Translating verbal descriptions involving rates and modified time intervals into algebraic expressions.

Alternative Method

Substitute a concrete number for hh to verify the translation. If h=10h = 10, half the hours is 5, and 3 hours less is 2 hours. At rate pp, energy generated is 2p2p. Substituting h=10h = 10 into E=p(12(10)−3)E = p\left(\frac{1}{2}(10) - 3\right) yields p(5−3)=2pp(5 - 3) = 2p, confirming the correct formula.
Estimated Time:1m 30s
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Topics

Basic Algebraic Expressions and One-Step Equations

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