Solving a Marathon Calorie and Budget Problem Using a System of Inequalities
Problem: Tenison needs to eat at least an extra 1{,}000 calories a day to prepare for a marathon. He has only $25 to spend on the extra food and will spend it on donuts (costing $0.75 and having calories each) and energy drinks (costing $2 and having calories each).
ⓐ Set up the system. Let = the number of donuts and = the number of energy drinks. Translating the two constraints:
- "At least 1{,}000 calories" →
- "No more than $25" →
Because quantities cannot be negative, we also have and . The system is:
ⓑ Graph the system. Graph as a solid boundary line. Testing (0, 0): is false, so shade the side away from the origin. Graph as a solid boundary line. Testing (0, 0): is true, so shade the side containing the origin. The solution is the doubly-shaded region in the first quadrant.
ⓒ To determine if donuts and energy drinks satisfy the needs, we test the point (8, 4) in the inequalities:
- Calories: (True)
- Budget: (True) Since both constraints are met, he can buy donuts and energy drinks.
ⓓ To determine if donut and energy drinks satisfy the needs, we test the point (1, 3):
- Calories: (False) Since the calorie constraint is not met, he cannot buy donut and energy drinks.
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Intermediate Algebra @ OpenStax
Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
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