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Zorluk: OrtaLinear Equations in Two Variables

The total daily cost CC, in dollars, for a custom apparel company to manufacture xx shirts is modeled by a linear equation. The table below shows the total daily cost for two different numbers of shirts manufactured:

Number of shirts, xxTotal daily cost, CC (dollars)
12260
20380

Based on the model, what is the total daily cost, in dollars, to manufacture 35 shirts?

  1. A
    262
  2. B
    536
  3. 605Cevap
  4. D
    785

Cevap

605
To find the total daily cost to manufacture 35 shirts, we first determine the linear relationship C=mx+bC = mx + b between the number of shirts manufactured, xx, and the total cost, CC. The slope mm is the constant rate of change, calculated as m=380−26020−12=1208=15m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15 dollars per shirt. Using the point-slope form with the point (12,260)(12, 260), we get C−260=15(x−12)C - 260 = 15(x - 12), which simplifies to C=15x+80C = 15x + 80. Substituting 3535 for xx in this equation yields C=15(35)+80=525+80=605C = 15(35) + 80 = 525 + 80 = 605. Therefore, the total daily cost to manufacture 35 shirts is 605 dollars.

Adım Adım Çözüm

1
Calculate the slope (rate of change) of the linear relationship using the two points from the table, (12,260)(12, 260) and (20,380)(20, 380).
Slope m=380−26020−12=1208=15m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15 dollars per shirt.
Since the relationship is linear, the rate of change is constant.
2
Find the equation of the line using the point-slope form C−C1=m(x−x1)C - C_1 = m(x - x_1) with the point (12,260)(12, 260) and slope m=15m = 15.
C−260=15(x−12)  ⟹  C=15x−180+260  ⟹  C=15x+80C - 260 = 15(x - 12) \implies C = 15x - 180 + 260 \implies C = 15x + 80.
This establishes the linear equation relating the number of shirts xx and the total daily cost CC.
3
Substitute x=35x = 35 into the linear equation to find the total daily cost to manufacture 35 shirts.
C=15(35)+80=525+80=605C = 15(35) + 80 = 525 + 80 = 605 dollars.
This evaluates the linear model at the desired quantity.

Anahtar Kavram

Determining a linear equation in two variables from a table of values and using it to predict a value.
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Konular

Linear Equations in Two Variables

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