This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Part (a): Let x be the number of dining chairs and y be the number of garden chairs. Step 1: Translate the given conditions into inequalities. • "at least 10 dining chairs": x 10 • "at least 20 garden chairs": y 20 • "not more than 80 chairs altogether": x + y 80 • "number of garden chairs must not be more than three times the number of dining chairs": y 3x The four inequalities are: x 10 y 20 x + y 80 y 3x Part (b): To draw the graph and shade the unwanted region: Step 1: Draw the x and y axes. Label the x-axis as "Number of Dining Chairs" and the y-axis as "Number of Garden Chairs". Step 2: Apply the scale: 2cm represents 10 chairs on both axes. This means that for every 2cm interval, mark 10, 20, 30, ..., up to 80. Step 3: Draw the boundary lines for each inequality: • For x 10, draw the vertical line x = 10. • For y 20, draw the horizontal line y = 20. • For x + y 80, draw the line x + y = 80. This line passes through (80, 0) and (0, 80). • For y 3x, draw the line y = 3x. This line passes through (0, 0), (10, 30), (20, 60), etc. Step 4: Identify the feasible region (the region satisfying all inequalities). • x 10: The region to the right of x = 10. • y 20: The region above y = 20. • x + y 80: The region below the line x + y = 80. • y 3x: The region below the line y = 3x. Step 5: Shade the unwanted region. This means shading the areas that do not satisfy the inequalities. • Shade the region to the left of x = 10. • Shade the region below y = 20. • Shade the region above x + y = 80. • Shade the region above y = 3x. The unshaded region will be the feasible region, which is a polygon with vertices at the intersections of these lines. The vertices of the feasible region are: • Intersection of x=10 and y=20: (10, 20) • Intersection of x=10 and y=3x: (10, 30) • Intersection of y=3x and x+y=80: x+3x=80 4x=80 x=20. Then y=3(20)=60. So, (20, 60) • Intersection of y=20 and x+y=80: x+20=80 x=60. So, (60, 20) Part (c): Step 1: Define the profit function. Profit on a dining chair = K80.00 Profit on a garden chair = K50.00 The total profit P is given by: P = 80x + 50y Step 2: Evaluate the profit function at each vertex of the feasible region to find the maximum profit. • At (10, 20): P = 80(10) + 50(20) = 800 + 1000 = 1800 • At (10, 30): P = 80(10) + 50(30) = 800 + 1500 = 2300 • At (20, 60): P = 80(20) + 50(60) = 1600 + 3000 = 4600 • At (60, 20): P = 80(60) + 50(20) = 4800 + 1000 = 5800 Step 3: Identify the combination of chairs that yields the maximum profit. The maximum profit occurs at (60, 20). Mipando should make 60 dining chairs and 20 garden chairs. Part (d): Step 1: State the maximum profit found in Part (c). The maximum profit is the highest value calculated from the vertices. The maximum profit is K5800.00. 3 done, 2 left today. You're making progress.