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
To find the maximum value of the objective function Z = 12x + 8y, we need to evaluate it at each vertex of the feasible region (the shaded area). Step 1: Identify the equations of the lines forming the boundaries of the feasible region. Line 1 passes through (20, 0) and (0, 16). The equation is (x)/(20) + (y)/(16) = 1. Multiplying by 80 (LCM of 20 and 16), we get 4x + 5y = 80. Line 2 passes through (10, 0) and (0, -8). The equation is (x)/(10) + (y)/(-8) = 1. Multiplying by 40 (LCM of 10 and 8), we get 4x - 5y = 40. Step 2: Identify the vertices of the feasible region from the graph. The vertices of the shaded triangular region are: 1. The y-intercept of Line 1: (0, 16) 2. The x-intercept of Line 2: (10, 0) 3. The intersection point of Line 1 and Line 2. Step 3: Find the intersection point of Line 1 and Line 2. We have the system of equations: 4x + 5y = 80 (1) 4x - 5y = 40 (2) Add equation (1) and equation (2): (4x + 5y) + (4x - 5y) = 80 + 40 8x = 120 x = (120)/(8) x = 15 Substitute x=15 into equation (1): 4(15) + 5y = 80 60 + 5y = 80 5y = 80 - 60 5y = 20 y = 4 So, the third vertex is (15, 4). Step 4: Evaluate the objective function Z = 12x + 8y at each vertex. 1. At (0, 16): Z = 12(0) + 8(16) = 0 + 128 = 128 2. At (10, 0): Z = 12(10) + 8(0) = 120 + 0 = 120 3. At (15, 4): Z = 12(15) + 8(4) = 180 + 32 = 212 Step 5: Determine the maximum value. Comparing the values 128, 120, and 212, the maximum value is 212. The final answer is D. 212. Just one more to go today! Send it over.