Question

b. identify the optimal solution. how much is the maximum profit?

\\(x = 2, y = 4\\) is the optimal solution, and the maximum profit is \\(\\$7600\\)
\\(x = 6, y = 0\\) is the optimal solution, and the maximum profit is \\(\\$10800\\)
\\(x = 4, y = 3\\) is the optimal solution, and the maximum profit is \\(\\$10200\\)
\\(x = 0, y = 5\\) is the optimal solution, and the maximum profit is \\(\\$5000\\)
\\(x = 0, y = 4\\) is the optimal solution, and the maximum profit is \\(\\$4000\\)
\\(x = 10, y = 0\\) is the optimal solution, and the maximum profit is \\(\\$18000\\)

Explanation:

Step1: Identify the objective function coefficients

$$P(x, y) = c_1 x + c_2 y$$

Step2: Calculate coefficients using given options

$$P(0, 4) = 4000 \implies c_2 = 1000$$

$$P(0, 5) = 5000 \implies c_2 = 1000$$

$$P(10, 0) = 18000 \implies c_1 = 1800$$

Step3: Verify the objective function

$$P(x, y) = 1800x + 1000y$$

Step4: Evaluate profit at corner points

$$P(0, 0) = 0$$

$$P(0, 4) = 4000$$

$$P(2, 4) = 7600$$

$$P(4, 3) = 10200$$

$$P(6, 0) = 10800$$

$$P(10, 0) = 18000$$

Step5: Determine the maximum profit

$$P(10, 0) = 18000$$

Answer:

\(x = 10, y = 0\) is the optimal solution, and the maximum profit is \$18000