7

Set 6: Linear Inequalities

Explanation

Answer: A

A nutritionist wants to mix two foods. Food A has 2g protein, 4g carbs. Food B has 3g protein, 2g carbs. Goal: At least 12g protein, at least 16g carbs. Minimize Calories (A: 50, B: 40). Which objective function and constraints apply?

A.

Min C=50A+40BC = 50A + 40B; 2A+3B≥12,4A+2B≥16,A,B≥02A + 3B \geq 12, 4A + 2B \geq 16, A,B \geq 0

✓ Correct
B.

Min C=50A+40BC = 50A + 40B; 2A+3B≤12,4A+2B≤16,A,B≥02A + 3B \leq 12, 4A + 2B \leq 16, A,B \geq 0

C.

Max C=50A+40BC = 50A + 40B; 2A+3B≥12,4A+2B≥16,A,B≥02A + 3B \geq 12, 4A + 2B \geq 16, A,B \geq 0

D.

Min C=2A+3BC = 2A + 3B; 50A+40B≥12,A,B≥050A + 40B \geq 12, A,B \geq 0

Detailed Explanation

Choice A is correct. Choice A is the correct answer. Match goals to functions and constraints. 1. Objective: Minimize Calories →Min C=50A+40B\rightarrow \text{Min } C = 50 A + 40 B 2. Protein: 2A+3B≥122A + 3 B \geq 12 (At least 12) 3. Carbs: 4A+2B≥164A + 2 B \geq 16 (At least 16) 4. Non-negativity: A,B≥0A, B \geq 0 Strategic Tip: "At least" means ≥\geq. Minimize means finding the lowest value of the objective function. Choice B is incorrect because ≤\leq sets a maximum limit, not a minimum requirement. Choice C is incorrect because it maximizes calories. Choice D is incorrect because it swaps the objective function with a constraint.

Key Steps:

•

The correct answer is Min C=50A+40BC = 50A + 40B; 2A+3B≥12,4A+2B≥16,A,B≥02A + 3B \geq 12, 4A + 2B \geq 16, A,B \geq 0

Why others are wrong:
B: Choice B is incorrect and may result from a calculation error.
C: Choice C is incorrect and may result from a calculation error.
D: Choice D is incorrect and may result from a calculation error.
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