Constraint Systems

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Constraint Systems · Grade 9 · Worksheet 3

1. Aisha is designing a rectangular stage for a school play. The stage must have a perimeter of at least 50 meters but no more than 70 meters. Additionally, the area of the stage must be at least 150 square meters. If the length is 5 meters more than twice the width, write a system of inequalities that represents all possible dimensions for Aisha's stage, using w for width and l for length. Answer: 2w + 2l ≥ 50, 2w + 2l ≤ 70, l = 2w + 5, l × w ≥ 150 Solution: Write the perimeter constraints. The perimeter P = 2w + 2l must be at least 50 and no more than 70, so: 2w + 2l ≥ 50 and 2w + 2l ≤ 70. Write the relationship between length and width.
   Full step-by-step solution

   Step 1: Write the perimeter constraints. The perimeter P = 2w + 2l must be at least 50 and no more than 70, so: 2w + 2l ≥ 50 and 2w + 2l ≤ 70. Step 2: Write the relationship between length and width. The length is 5 more than twice the width, so: l = 2w + 5. Step 3: Write the area constraint. The area A = l × w must be at least 150, so: l × w ≥ 150. Step 4: Combine all constraints into a system: 2w + 2l ≥ 50, 2w + 2l ≤ 70, l = 2w + 5, l × w ≥ 150.

2. Aroha is organizing a school event with a budget of $320. She needs to buy chairs that cost $18 each and tables that cost $25 each. She must have at least 12 chairs and at least 8 tables, and the total number of items must be at least 25. Write the system of inequalities that represents these constraints, where x is the number of chairs and y is the number of tables. Answer: 18x + 25y ≤ 320, x ≥ 12, y ≥ 8, x + y ≥ 25 Solution: Write the cost constraint. Chairs cost $18 each and tables cost $25 each, with a total budget of $320. So 18x + 25y ≤ 320.
   Full step-by-step solution

   Step 1: Write the cost constraint. Chairs cost $18 each and tables cost $25 each, with a total budget of $320. So 18x + 25y ≤ 320. Step 2: Write the minimum chairs constraint. She needs at least 12 chairs, so x ≥ 12. Step 3: Write the minimum tables constraint. She needs at least 8 tables, so y ≥ 8. Step 4: Write the total items constraint. She needs at least 25 items total, so x + y ≥ 25. Step 5: Combine all constraints: 18x + 25y ≤ 320, x ≥ 12, y ≥ 8, x + y ≥ 25.

3. Matiu is buying school supplies with a budget of $85. He needs at least 15 notebooks and pens combined. Notebooks cost $4 each and pens cost $2 each. He needs at least twice as many pens as notebooks. Write the system of inequalities representing these constraints. Answer: 4x + 2y ≤ 85, x + y ≥ 15, y ≥ 2x, x ≥ 0, y ≥ 0 Solution: Let x = number of notebooks, y = number of pens Budget constraint: 4x + 2y ≤ 85 (notebooks cost $4 each, pens cost $2 each, total ≤ $85) Minimum items constraint: x + y ≥ 15 (at least 15 items total) Ratio constraint: y ≥ 2x (at least twice as many pens as notebooks) Non-negativity constraints:…
   Full step-by-step solution

   Step 1: Let x = number of notebooks, y = number of pens Step 2: Budget constraint: 4x + 2y ≤ 85 (notebooks cost $4 each, pens cost $2 each, total ≤ $85) Step 3: Minimum items constraint: x + y ≥ 15 (at least 15 items total) Step 4: Ratio constraint: y ≥ 2x (at least twice as many pens as notebooks) Step 5: Non-negativity constraints: x ≥ 0, y ≥ 0 (cannot buy negative items) Step 6: The complete system is: 4x + 2y ≤ 85, x + y ≥ 15, y ≥ 2x, x ≥ 0, y ≥ 0

4. Aroha is organizing a school fundraiser where she sells two types of gift baskets: small and large. Each small basket requires 3 hours of labor and 2 square meters of storage space. Each large basket requires 5 hours of labor and 7 square meters of storage space. Aroha has at most 45 hours of labor available and at most 63 square meters of storage space. She must prepare at least 3 small baskets and at least 5 large baskets to meet demand. Write a system of inequalities that represents all possible numbers of small baskets (x) and large baskets (y) Aroha can prepare. Answer: 3x + 5y ≤ 45, 2x + 7y ≤ 63, x ≥ 3, y ≥ 5 Solution: Let x = number of small baskets, y = number of large baskets. Labor constraint: each small basket uses 3 hours, each large uses 5 hours, total hours ≤ 45 → 3x + 5y ≤ 45.
   Full step-by-step solution

   Step 1: Let x = number of small baskets, y = number of large baskets. Step 2: Labor constraint: each small basket uses 3 hours, each large uses 5 hours, total hours ≤ 45 → 3x + 5y ≤ 45. Step 3: Storage constraint: each small basket uses 2 m², each large uses 7 m², total space ≤ 63 → 2x + 7y ≤ 63. Step 4: Minimum small baskets: x ≥ 3. Step 5: Minimum large baskets: y ≥ 5. Step 6: Combine all inequalities: 3x + 5y ≤ 45, 2x + 7y ≤ 63, x ≥ 3, y ≥ 5. This system represents all possible combinations (x, y) that satisfy Aroha's constraints.

5. Mason is organizing a school fundraiser where he will sell two types of tickets: adult tickets at $8 each and student tickets at $5 each. He must sell at least 150 tickets in total to cover the venue cost. Additionally, the total revenue from ticket sales must be at least $900 to meet the fundraising goal. However, due to seating capacity, the number of student tickets cannot exceed twice the number of adult tickets. Write a system of inequalities that represents these constraints, using x for the number of adult tickets and y for the number of student tickets. Answer: x + y ≥ 150, 8x + 5y ≥ 900, y ≤ 2x, x ≥ 0, y ≥ 0 Solution: Represent the total ticket constraint. Mason needs at least 150 tickets, so x + y ≥ 150. Represent the revenue constraint.
   Full step-by-step solution

   Step 1: Represent the total ticket constraint. Mason needs at least 150 tickets, so x + y ≥ 150. Step 2: Represent the revenue constraint. Each adult ticket is $8 and each student ticket is $5, so total revenue is 8x + 5y. This must be at least $900, so 8x + 5y ≥ 900. Step 3: Represent the seating capacity constraint. The number of student tickets cannot exceed twice the number of adult tickets, so y ≤ 2x. Step 4: Represent non-negativity constraints. Since you cannot sell negative tickets, x ≥ 0 and y ≥ 0. Step 5: Combine all constraints into a system: x + y ≥ 150 8x + 5y ≥ 900 y ≤ 2x x ≥ 0 y ≥ 0 This system represents all possible numbers of adult and student tickets Mason can sell.

6. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(6,8). A circle is inscribed inside this triangle such that it is tangent to all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: A(0,0) B(6,0) C(6,8) AB is along the x-axis from (0,0) to (6,0) → length = 6 BC is vertical from (6,0) to (6,8) → length = 8 AC is the hypotenuse from (0,0) to (6,8).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A(0,0) B(6,0) C(6,8) So: AB is along the x-axis from (0,0) to (6,0) → length = 6 BC is vertical from (6,0) to (6,8) → length = 8 AC is the hypotenuse from (0,0) to (6,8). --- **Step 2: Find side lengths** AB = 6 BC = 8 AC = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 So triangle sides: 6, 8, 10. --- **Step 3: Formula for inradius of a right triangle** For a right triangle with legs a and b, hypotenuse c, the inradius r is: r = (a + b - c) / 2 --- **Step 4: Apply formula** a = 6, b = 8, c = 10 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. --- **Final answer:** 2