Optimization of a Production Process?

📌 CONCEPT: The problem requires finding the optimal diameter of a metal rod, given a fixed length, to minimize the production cost, which is directly proportional to the surface area of the rod.

📐 RULE / FORMULA: The formula to calculate the surface area of a cylinder is given by A = 2πrh, where r is the radius and h is the height of the cylinder. Since the cost is directly proportional to the surface area, we can write the cost function as C = kA = k(2πrh).

💡 WORKED EXAMPLE: Given a rod of length 1 meter, we need to find the optimal diameter that minimizes the cost. Let's assume the radius is r. The surface area A = 2πrh = 2πr(1) = 2πr. Now, the cost function becomes C = k(2πr). To minimize the cost, we take the derivative of C with respect to r and set it equal to zero: dC/dr = k(2π) = 0. Solving for r, we get r = 0, which is not a feasible solution. However, we can see that the cost function is a decreasing function of r, meaning that the cost decreases as the radius increases. Therefore, the optimal diameter is infinite, but since it's not feasible, we can say that the optimal diameter is a very large value.

⚠️ COMMON MISTAKE: Students often make the mistake of assuming that the optimal diameter is a fixed value, whereas in reality, it depends on the cost function and the constraints given in the problem.