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Zorluk: OrtaAlgebraic Word Problems and Modeling

A clean energy technology company manufactures two models of solar panels: Model X and Model Y. Model X produces 150150 kilowatt-hours (kWh) of electricity per day and costs $400\$400 to manufacture, while Model Y produces 200200 kWh of electricity per day and costs $550\$550 to manufacture. A solar energy project purchased a total of 5050 panels for a total manufacturing cost of $24,500\$24,500. What is the total daily electricity production, in kilowatt-hours, of all 5050 panels combined?

  1. A
    8,5008,500
  2. B
    8,7508,750
  3. 9,0009,000Cevap
  4. D
    9,5009,500
  5. E
    10,00010,000

Cevap

The total daily electricity production of all 5050 panels combined is 9,0009,000 kWh.
The correct option is 9,0009,000 kWh. Defining xx as the number of Model X panels and yy as the number of Model Y panels gives the equations x+y=50x + y = 50 and 400x+550y=24,500400x + 550y = 24,500. Substituting y=50−xy = 50 - x yields 400x+550(50−x)=24,500400x + 550(50 - x) = 24,500, which simplifies to −150x=−3,000-150x = -3,000, so x=20x = 20 and y=30y = 30. Computing the total output yields 20(150)+30(200)=3,000+6,000=9,00020(150) + 30(200) = 3,000 + 6,000 = 9,000 kWh.

Adım Adım Çözüm

1
Define variables for the quantities of each panel model.
Let xx represent the number of Model X panels and yy represent the number of Model Y panels.
Establishing explicit variables allows formulating a system of linear equations from the word problem.
2
Set up equations for total panel quantity and total manufacturing cost.
System equations: x+y=50x + y = 50 and 400x+550y=24,500400x + 550y = 24,500.
The total number of panels is 5050, and the combined manufacturing cost equals $24,500\$24,500.
3
Solve the system using substitution.
Substitute y=50−xy = 50 - x into the cost equation: 400x+550(50−x)=24,500  ⟹  400x+27,500−550x=24,500  ⟹  −150x=−3,000  ⟹  x=20400x + 550(50 - x) = 24,500 \implies 400x + 27,500 - 550x = 24,500 \implies -150x = -3,000 \implies x = 20. Consequently, y=50−20=30y = 50 - 20 = 30.
Determining x=20x = 20 and y=30y = 30 gives the exact number of Model X and Model Y panels purchased.
4
Calculate the total daily electricity production.
Total production =20(150)+30(200)=3,000+6,000=9,000= 20(150) + 30(200) = 3,000 + 6,000 = 9,000 kWh.
Multiply the quantity of each panel model by its respective daily output rate and sum the products.

Anahtar Kavram

Formulating and solving a system of two linear equations from a real-world scenario to find unknown quantities and evaluate a secondary combination function.
Tahmini Süre:1m 30s
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Algebraic Word Problems and Modeling

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