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Difficulty: HardTranslating and Solving Algebraic Word Problems

A company produces solar-powered chargers. The daily production cost, in dollars, is a linear function of the number of chargers produced. The setup cost is the cost when 00 chargers are produced. If the company produces 1515 chargers, the total daily cost is 400400 dollars. If the company produces 2525 chargers, the total daily cost is 580580 dollars. The company updates its production process, which reduces the cost per charger by 20%20\% but increases the setup cost by 5050 dollars. Under the updated process, what is the total daily cost, in dollars, to produce 3030 chargers?

  1. A
    562562 dollars
  2. B
    586586 dollars
  3. 612612 dollarsAnswer
  4. D
    714714 dollars
  5. E
    828828 dollars

Answer

The correct daily cost under the updated process is 612612 dollars.
The correct answer is 612612 dollars. First, the relationship between unit cost and setup cost is modeled as a linear equation. Subtracting the cost of producing 1515 chargers (400400 dollars) from the cost of producing 2525 chargers (580580 dollars) gives the cost of producing the 1010 additional units, which is 180180 dollars. This determines the unit cost is 1818 dollars per charger. Substituting this value back shows the original setup cost is 130130 dollars. The updated rate decreases the unit cost to 14.4014.40 dollars (80%80\% of 1818) and increases the setup cost to 180180 dollars (130+50130 + 50). For 3030 chargers, the total daily cost is 14.40(30)+180=61214.40(30) + 180 = 612 dollars.

Step-by-Step Solution

1
Set up a system of linear equations using the cost function C(c)=mc+SC(c) = mc + S, where mm is the cost per charger, SS is the setup cost, and cc is the number of chargers.
Equation 1: 15m+S=40015m + S = 400
Equation 2: 25m+S=58025m + S = 580
This establishes the relationship between production volume and total cost under the initial process.
2
Solve the system of equations for the unit cost mm and the setup cost SS.
Subtracting Equation 1 from Equation 2 yields 10m=180  ⟹  m=1810m = 180 \implies m = 18.
Substituting m=18m = 18 into Equation 1 yields 15(18)+S=400  ⟹  270+S=400  ⟹  S=13015(18) + S = 400 \implies 270 + S = 400 \implies S = 130.
This determines the original pricing parameters.
3
Apply the updates to the cost parameters.
New unit cost: 18×(1−0.20)=14.4018 \times (1 - 0.20) = 14.40 dollars.
New setup cost: 130+50=180130 + 50 = 180 dollars.
This accounts for the 20%20\% decrease in the variable cost and the 5050 dollar increase in the fixed setup cost.
4
Evaluate the new linear cost function for 3030 chargers.
Cnew(30)=14.40(30)+180=432+180=612C_{\text{new}}(30) = 14.40(30) + 180 = 432 + 180 = 612 dollars.
This calculates the total daily cost under the updated process.

Key Concept

Translating and Solving Algebraic Word Problems

Alternative Method

Instead of solving for the setup cost first, note that under the original process, the cost of 3030 chargers would be the cost of 1515 chargers plus the cost of 1515 more units: 400+15(18)=670400 + 15(18) = 670 dollars. The updated process reduces the rate of each of the 3030 units by 20%20\% of 1818 dollars (saving 3.60×30=1083.60 \times 30 = 108 dollars) and increases the setup cost by 5050 dollars. Thus, the new cost is 670−108+50=612670 - 108 + 50 = 612 dollars.
Estimated Time:2m 0s
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Topics

Translating and Solving Algebraic Word Problems

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