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Zorluk: Çok zorAlgebraic Word Problems and Modeling

A logistics company calculates the shipping cost CC, in dollars, for a parcel of weight ww pounds shipped over a distance of dd miles using the following piecewise pricing model:
- A fixed base fee of $10\$10.
- An additional weight surcharge of $1.50\$1.50 per pound for every pound (or fraction thereof) exceeding 55 pounds.
- An additional distance surcharge of $0.05\$0.05 per mile for every mile exceeding 100100 miles.
- A flat combined heavy-distance surcharge of $25\$25 applied if and only if w>10w > 10 AND d>500d > 500.

A customer ships two parcels, Parcel 1 and Parcel 2:
- Parcel 1 weighs w1w_1 pounds (where w1w_1 is an integer greater than 10) and is shipped a distance of d1d_1 miles (where d1d_1 is an integer greater than 500).
- Parcel 2 weighs w2=w1−4w_2 = w_1 - 4 pounds and is shipped a distance of d2=d1+200d_2 = d_1 + 200 miles.

Let C1C_1 be the shipping cost of Parcel 1 and C2C_2 be the shipping cost of Parcel 2.

Which of the following statements MUST be true? Select all such statements.

  1. If w1=12w_1 = 12, then Parcel 1 costs exactly $21\$21 more to ship than Parcel 2.Cevap
  2. If w1>14w_1 > 14, Parcel 2 always costs exactly $4\$4 more to ship than Parcel 1.Cevap
  3. C
    The total combined shipping cost of both parcels, C1+C2C_1 + C_2, is independent of the distance d1d_1.
  4. D
    Parcel 2 always costs more to ship than Parcel 1, regardless of the value of w1w_1.
  5. E
    If w1=15w_1 = 15, the shipping cost of Parcel 2 is $26.50\$26.50 plus 0.05d10.05 d_1.

Cevap

The correct statements are the ones indicating that if the weight of Parcel 1 is 12 pounds, Parcel 1 costs exactly 21morethanParcel2,andiftheweightofParcel1isgreaterthan14pounds,Parcel2costsexactly21 more than Parcel 2, and if the weight of Parcel 1 is greater than 14 pounds, Parcel 2 costs exactly 4 more than Parcel 1.
The cost functions must be broken down by conditional threshold. For Parcel 1, both weight (>10) and distance (>500) thresholds are met, giving C_1 = 1.5w_1 + 0.05d_1 + 22.5. For Parcel 2, d_2 = d_1 + 200 > 700 is always > 500, but w_2 = w_1 - 4 exceeds 10 if and only if w_1 > 14. If w_1 = 12, w_2 = 8, so Parcel 2 gets no 25surcharge,resultinginC1−C2=21.Ifw1>14,w2>10,soParcel2getsthe25 surcharge, resulting in C_1 - C_2 = 21. If w_1 > 14, w_2 > 10, so Parcel 2 gets the 25 surcharge, giving C_2 - C_1 = 4.

Adım Adım Çözüm

1
Formulate the algebraic cost function C(w, d) for any parcel.
C(w, d) = 10 + 1.50(w - 5) + 0.05(d - 100) + S(w, d), where S(w, d) = 25 if w > 10 and d > 500, else S(w, d) = 0.
This establishes the piecewise linear equations governing shipping costs.
2
Express C_1 for Parcel 1 given w_1 > 10 and d_1 > 500.
Since w_1 > 10 and d_1 > 500, S(w_1, d_1) = 25. Thus, C_1 = 10 + 1.50(w_1 - 5) + 0.05(d_1 - 100) + 25 = 1.50 w_1 + 0.05 d_1 + 22.50.
Parcel 1 always meets both conditions for the $25 heavy-distance surcharge.
3
Analyze Parcel 2 when 10 < w_1 <= 14 (meaning 6 < w_2 <= 10).
For w_2 <= 10, S(w_2, d_2) = 0 even though d_2 > 500. C_2 = 10 + 1.50(w_1 - 9) + 0.05(d_1 + 100) + 0 = 1.50 w_1 + 0.05 d_1 + 1.50. Therefore, C_1 - C_2 = (1.50 w_1 + 0.05 d_1 + 22.50) - (1.50 w_1 + 0.05 d_1 + 1.50) = $21.00.
This confirms that for w_1 = 12, Parcel 1 costs exactly $21 more than Parcel 2.
4
Analyze Parcel 2 when w_1 > 14 (meaning w_2 > 10).
For w_2 > 10 and d_2 > 700 > 500, S(w_2, d_2) = 25. Thus, C_2 = 1.50 w_1 + 0.05 d_1 + 26.50. Subtracting C_1 gives C_2 - C_1 = (1.50 w_1 + 0.05 d_1 + 26.50) - (1.50 w_1 + 0.05 d_1 + 22.50) = $4.00.
This proves Parcel 2 costs exactly $4 more than Parcel 1 whenever w_1 > 14.

Anahtar Kavram

Piecewise linear modeling and conditional surcharge threshold evaluations
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Konular

Algebraic Word Problems and Modeling

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