Constraint Systems

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Answer Key & Explanations

Constraint Systems · Grade 9 · Worksheet 1

1. A company produces two models of laptops: Model A and Model B. The production facility has these constraints: The total number of laptops produced daily cannot exceed 200. At least 40 Model A laptops must be made to fulfill contracts. Due to component supply, Model B production cannot exceed 150 units. The profit is $80 for each Model A and $120 for each Model B. How many Model B laptops should be produced daily to maximize profit while satisfying all constraints? Answer: 150 Solution: Define variables - Let x = number of Model A laptops, y = number of Model B laptops - Total production: x + y ≤ 200 - Minimum Model A: x ≥ 40 - Maximum Model B: y ≤ 150 - Non-negative: x ≥ 0, y ≥ 0 Write the profit function: P = 80x + 120y Since Model B has higher profit per unit ($120 vs $80),…
   Full step-by-step solution

   Step 1: Define variables - Let x = number of Model A laptops, y = number of Model B laptops Step 2: Write the constraints as inequalities: - Total production: x + y ≤ 200 - Minimum Model A: x ≥ 40 - Maximum Model B: y ≤ 150 - Non-negative: x ≥ 0, y ≥ 0 Step 3: Write the profit function: P = 80x + 120y Step 4: Since Model B has higher profit per unit ($120 vs $80), we want to produce as many Model B as possible. Step 5: Check constraints: The maximum allowed for Model B is 150. Step 6: If y = 150, then x + 150 ≤ 200, so x ≤ 50. This satisfies x ≥ 40. Step 7: Verify all constraints are satisfied: x = 40-50, y = 150 gives x + y = 190-200 ≤ 200, x ≥ 40, y ≤ 150. Step 8: Calculate profit with maximum Model B: P = 80x + 120(150) = 80x + 18000. To maximize, use smallest x allowed (x = 40): P = 80(40) + 18000 = 3200 + 18000 = 21200. Step 9: Any other combination would give lower profit since Model B has higher profit margin. The answer is 150.

2. Lena is designing a rectangular garden with a perimeter of 40 meters. She wants the length to be at least 3 meters more than the width, and the area must be at least 60 square meters. Write a system of inequalities that represents the constraints on the garden's dimensions, using l for length and w for width. Answer: 2l + 2w = 40, l ≥ w + 3, l * w ≥ 60 Solution: - Perimeter = 40 meters - Length = l, Width = w - Length at least 3 meters more than width → l ≥ w + 3 - Area at least 60 square meters → l × w ≥ 60 We need a system of inequalities (and one equation) for l and w.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the problem** We have a rectangular garden with: - Perimeter = 40 meters - Length = l, Width = w - Length at least 3 meters more than width → l ≥ w + 3 - Area at least 60 square meters → l × w ≥ 60 We need a system of inequalities (and one equation) for l and w. --- **Step 2: Perimeter equation** Perimeter of a rectangle = 2 × length + 2 × width So: 2l + 2w = 40 We can leave it as is, or simplify by dividing by 2: l + w = 20 But the given correct answer keeps it as 2l + 2w = 40, so we'll use that. --- **Step 3: Length and width relationship** "Length to be at least 3 meters more than width" means: l ≥ w + 3 --- **Step 4: Area condition** Area = l × w "Area must be at least 60" means: l × w ≥ 60 --- **Step 5: Combine all constraints** From above: 1. Perimeter: 2l + 2w = 40 2. Length condition: l ≥ w + 3 3. Area condition: l × w ≥ 60 --- **Final system of inequalities (and one equation):** 2l + 2w = 40 l ≥ w + 3 l × w ≥ 60 --- This matches the given correct answer.

3. Mason is helping his family plan a rectangular vegetable garden. The garden must have a perimeter of no more than 28 meters. To ensure enough sunlight, the length must be at least 2 meters greater than the width. Also, to fit the vegetables they want to plant, the area must be at least 15 square meters. Write a system of inequalities that represents these constraints, using L for length and W for width. Answer: 2L + 2W ≤ 28, L ≥ W + 2, L × W ≥ 15 Solution: The perimeter of a rectangle is 2L + 2W. The problem says the perimeter is no more than 28 meters, so 2L + 2W ≤ 28. The length must be at least 2 meters greater than the width.
   Full step-by-step solution

   Step 1: The perimeter of a rectangle is 2L + 2W. The problem says the perimeter is no more than 28 meters, so 2L + 2W ≤ 28. Step 2: The length must be at least 2 meters greater than the width. This means L ≥ W + 2. Step 3: The area must be at least 15 square meters. Area of a rectangle is L × W, so L × W ≥ 15. Step 4: Combine all three constraints: 2L + 2W ≤ 28, L ≥ W + 2, L × W ≥ 15. Final Answer: 2L + 2W ≤ 28, L ≥ W + 2, L × W ≥ 15

4. Hana is organizing a school fundraiser and plans to sell two types of gift baskets: small and large. Each small basket requires 3 hours of labor and $20 in materials, while each large basket requires 5 hours of labor and $35 in materials. Hana has at most 120 hours of labor available and a budget of at most $700 for materials. She also wants to make at least 10 small baskets and at least 5 large baskets. Write a system of inequalities that represents all the constraints on the number of small baskets (x) and large baskets (y) Hana can produce. Answer: 3x + 5y ≤ 120, 20x + 35y ≤ 700, x ≥ 10, y ≥ 5, x ≥ 0, y ≥ 0 Solution: Define variables: let x = number of small baskets, y = number of large baskets. Labor constraint: each small basket takes 3 hours, each large takes 5 hours, total ≤ 120 hours → 3x + 5y ≤ 120.
   Full step-by-step solution

   Step 1: Define variables: let x = number of small baskets, y = number of large baskets. Step 2: Labor constraint: each small basket takes 3 hours, each large takes 5 hours, total ≤ 120 hours → 3x + 5y ≤ 120. Step 3: Material cost constraint: each small costs $20, each large costs $35, total ≤ $700 → 20x + 35y ≤ 700. Step 4: Minimum basket constraints: at least 10 small → x ≥ 10; at least 5 large → y ≥ 5. Step 5: Non-negativity constraints: x ≥ 0, y ≥ 0 (already covered by minimums, but included for completeness). Final system: 3x + 5y ≤ 120, 20x + 35y ≤ 700, x ≥ 10, y ≥ 5, x ≥ 0, y ≥ 0.

5. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). A circle is inscribed inside this triangle such that it touches all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: We have a right triangle with vertices A(0,0), B(6,0), and C(0,8). The circle is inscribed inside the triangle, meaning it touches all three sides. This is called the incircle.
   Full step-by-step solution

   Step 1: Understand the problem We have a right triangle with vertices A(0,0), B(6,0), and C(0,8). The circle is inscribed inside the triangle, meaning it touches all three sides. This is called the incircle. We need to find the radius of the incircle. Step 2: Recall the formula for the inradius of a right triangle For a right triangle with legs a and b and hypotenuse c, the inradius r is given by: r = (a + b - c) / 2 where c = sqrt(a^2 + b^2). Step 3: Identify the side lengths From the coordinates: Side AB is from (0,0) to (6,0), length a = 6. Side AC is from (0,0) to (0,8), length b = 8. Side BC is from (6,0) to (0,8), length c = sqrt((6-0)^2 + (0-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. So: a = 6, b = 8, c = 10. Step 4: Apply the formula r = (a + b - c) / 2 r = (6 + 8 - 10) / 2 r = (14 - 10) / 2 r = 4 / 2 r = 2. Step 5: Conclusion The radius of the inscribed circle is 2. We can also check: Area of triangle = (1/2) * base * height = (1/2) * 6 * 8 = 24. Semi-perimeter s = (6 + 8 + 10) / 2 = 24 / 2 = 12. Inradius from area formula: r = Area / s = 24 / 12 = 2. Both methods match.

6. Matiu is organizing a school fundraiser with a budget of $180. He needs to buy at least 15 items total. T-shirts cost $12 each and hats cost $8 each. Write the system of inequalities that represents these constraints, where x is the number of t-shirts and y is the number of hats. Answer: 12x + 8y ≤ 180, x + y ≥ 15, x ≥ 0, y ≥ 0 Solution: Write the budget constraint: Cost of t-shirts (12x) plus cost of hats (8y) must be ≤ 180 → 12x + 8y ≤ 180 Write the minimum quantity constraint: Total items (x + y) must be ≥ 15 → x + y ≥ 15 Write the non-negativity constraints: Cannot have negative t-shirts or hats → x ≥ 0, y ≥ 0 Combine all…
   Full step-by-step solution

   Step 1: Write the budget constraint: Cost of t-shirts (12x) plus cost of hats (8y) must be ≤ 180 → 12x + 8y ≤ 180 Step 2: Write the minimum quantity constraint: Total items (x + y) must be ≥ 15 → x + y ≥ 15 Step 3: Write the non-negativity constraints: Cannot have negative t-shirts or hats → x ≥ 0, y ≥ 0 Step 4: Combine all constraints: 12x + 8y ≤ 180, x + y ≥ 15, x ≥ 0, y ≥ 0 The system of inequalities is: 12x + 8y ≤ 180, x + y ≥ 15, x ≥ 0, y ≥ 0