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Difficulty: MediumAlgebraic Word Problems and Modeling

A coffee merchant creates a 5050-pound custom mixture combining Grade A coffee beans, which cost $8\$8 per pound, and Grade B coffee beans, which cost $14\$14 per pound. If the total cost of the mixture must be at least $460\$460 and at most $520\$520, which of the following could be the weight, in pounds, of Grade A coffee beans used in the mixture? Select all such weights.

  1. A
    2525
  2. 3232Answer
  3. 3535Answer
  4. 4040Answer
  5. E
    4545

Answer

The weight of Grade A coffee beans can be 32 pounds, 35 pounds, or 40 pounds.
Let xx represent the number of pounds of Grade A coffee beans. The remaining 50−x50 - x pounds consist of Grade B beans. The total cost of the mixture is given by 8x+14(50−x)=700−6x8x + 14(50 - x) = 700 - 6x. Setting up the given inequality constraints, we have 460≤700−6x≤520460 \le 700 - 6x \le 520. Subtracting 700 yields −240≤−6x≤−180-240 \le -6x \le -180. Dividing by −6-6 and reversing the inequality signs gives 30≤x≤4030 \le x \le 40. Therefore, any weight of Grade A beans between 30 and 40 pounds inclusive is valid. The values 32, 35, and 40 satisfy this condition.

Step-by-Step Solution

1
Define variables for the quantities of Grade A and Grade B beans.
Let xx be the weight in pounds of Grade A beans. Then 50−x50 - x is the weight in pounds of Grade B beans.
The total weight of the mixture is fixed at 50 pounds.
2
Set up an expression for the total cost of the mixture in terms of xx.
Total Cost = 8x+14(50−x)=700−6x8x + 14(50 - x) = 700 - 6x.
Multiply the weight of each component by its price per pound.
3
Set up the compound inequality representing the given cost constraints.
460≤700−6x≤520460 \le 700 - 6x \le 520.
The total cost must be at least $460\$460 and at most $520\$520.
4
Solve the compound inequality for xx.
Subtracting 700 from all parts gives −240≤−6x≤−180-240 \le -6x \le -180. Dividing by −6-6 and reversing the inequality signs yields 30≤x≤4030 \le x \le 40.
Dividing by a negative number reverses the direction of the inequality signs.
5
Identify which options fall within the valid range [30,40][30, 40].
The values 32, 35, and 40 fall within the range 30≤x≤4030 \le x \le 40.
Any weight between 30 and 40 pounds inclusive satisfies the cost constraint.

Key Concept

Linear Modeling and Inequality Constraints in Mixture Problems
Estimated Time:1m 40s
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Topics

Algebraic Word Problems and Modeling

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