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Difficulty: MediumAlgebraic Word Problems and Modeling

A courier service calculates the shipping cost for a package based on its weight ww, in pounds. For packages weighing up to 2020 pounds, the cost is a flat fee of $12.00\$12.00 plus $2.50\$2.50 per pound. For packages weighing more than 2020 pounds, the cost is a flat fee of $20.00\$20.00 plus $2.00\$2.00 per pound. If the total shipping cost for a package was strictly greater than $45.00\$45.00 and at most $70.00\$70.00, which of the following could be the weight of the package, in pounds? Select all such weights.

  1. A
    1212 pounds
  2. 1515 poundsAnswer
  3. 2020 poundsAnswer
  4. 2424 poundsAnswer
  5. E
    2626 pounds

Answer

The valid weights of the package are 15 pounds, 20 pounds, and 24 pounds.
The weight of the package must result in a total cost C(w)C(w) such that $45.00<C(w)≤$70.00\$45.00 < C(w) \le \$70.00. Evaluating the options:
- For 15 pounds: 12+2.50(15)=$49.5012 + 2.50(15) = \$49.50, which falls within the range.
- For 20 pounds: 12+2.50(20)=$62.0012 + 2.50(20) = \$62.00, which falls within the range.
- For 24 pounds: 20+2.00(24)=$68.0020 + 2.00(24) = \$68.00, which falls within the range.

Step-by-Step Solution

1
Formulate the cost function C(w)C(w) as a piecewise model.
C(w)=12+2.50wC(w) = 12 + 2.50w for 0<w≤200 < w \le 20, and C(w)=20+2.00wC(w) = 20 + 2.00w for w>20w > 20.
The cost structure changes depending on whether the weight exceeds 2020 pounds.
2
Set up and solve the inequality 45<C(w)≤7045 < C(w) \le 70 for packages up to 2020 pounds.
45<12+2.50w≤70  ⟹  33<2.50w≤58  ⟹  13.2<w≤23.245 < 12 + 2.50w \le 70 \implies 33 < 2.50w \le 58 \implies 13.2 < w \le 23.2. Restricting to w≤20w \le 20 gives 13.2<w≤2013.2 < w \le 20.
This determines the valid weight interval for packages billed under the first tier.
3
Set up and solve the inequality 45<C(w)≤7045 < C(w) \le 70 for packages over 2020 pounds.
45<20+2.00w≤70  ⟹  25<2.00w≤50  ⟹  12.5<w≤2545 < 20 + 2.00w \le 70 \implies 25 < 2.00w \le 50 \implies 12.5 < w \le 25. Restricting to w>20w > 20 gives 20<w≤2520 < w \le 25.
This determines the valid weight interval for packages billed under the second tier.
4
Combine the valid intervals and evaluate each option.
The overall valid weight interval is 13.2<w≤2513.2 < w \le 25. Testing the values: 12 pounds is outside the range; 15 pounds, 20 pounds, and 24 pounds are within the range; 26 pounds is outside the range.
Any weight strictly greater than 13.213.2 pounds and up to 2525 pounds produces a cost between $45.00\$45.00 and $70.00\$70.00.

Key Concept

Algebraic Modeling of Piecewise Functions and Compound Inequalities
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Topics

Algebraic Word Problems and Modeling

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