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
Step 1: Define the variables and calculate the expected values of sales. Let X be the number of tubs sold. The selling price per tub is €2. The revenue function is R(x) = 2x. The probability distribution for X is given as: x_i: 100, 200, 300, 400, 500 P(x_i): 0.15, 0.20, 0.25, 0.28, 0.12 First, calculate the expected number of tubs sold, E[X]: E[X] = x_i P(x_i) E[X] = (100 × 0.15) + (200 × 0.20) + (300 × 0.25) + (400 × 0.28) + (500 × 0.12) E[X] = 15 + 40 + 75 + 112 + 60 E[X] = 302 tubs Next, calculate the expected value of X^2, E[X^2], as some cost functions depend on x^2: E[X^2] = x_i^2 P(x_i) E[X^2] = (100^2 × 0.15) + (200^2 × 0.20) + (300^2 × 0.25) + (400^2 × 0.28) + (500^2 × 0.12) E[X^2] = (10000 × 0.15) + (40000 × 0.20) + (90000 × 0.25) + (160000 × 0.28) + (250000 × 0.12) E[X^2] = 1500 + 8000 + 22500 + 44800 + 30000 E[X^2] = 106800 The expected revenue is E[R(x)] = E[2x] = 2E[X]: E[R(x)] = 2 × 302 = 604 € Step 2: Calculate the expected profit for each procedure. The profit G(x) for each procedure is G(x) = R(x) - C(x), where C(x) is the cost function. We need to find E[G(x)] = E[R(x)] - E[C(x)]. Procedure A: Cost function: C_A(x) = 0.3x + 400 Expected cost: E[C_A(x)] = 0.3 E[X] + 400 E[C_A(x)] = (0.3 × 302) + 400 = 90.6 + 400 = 490.6 € Expected profit: E[G_A(x)] = E[R(x)] - E[C_A(x)] = 604 - 490.6 = 113.4 € Procedure B: Cost function: C_B(x) = 1x + 0.04\% x^2 = x + 0.0004x^2 Expected cost: E[C_B(x)] = E[X] + 0.0004 E[X^2] E[C_B(x)] = 302 + (0.0004 × 106800) = 302 + 42.72 = 344.72 € Expected profit: E[G_B(x)] = E[R(x)] - E[C_B(x)] = 604 - 344.72 = 259.28 € Procedure C: Cost function: C_C(x) = 0.5x + 0.06\% x^2 + 150 = 0.5x + 0.0006x^2 + 150 Expected cost: E[C_C(x)] =