Cost Minimization Problem?

📌 CONCEPT: The cost minimization problem is a technique used in optimization where we find the dimensions of a shape that minimize a certain cost or expense, given some constraints.

📐 RULE / FORMULA: The cost is directly proportional to the length of the wire, which can be expressed as C = kL, where C is the cost, k is the constant of proportionality, and L is the length of the wire. For a rectangle with length 'l' and breadth 'b', the perimeter P is given by P = 2(l + b), and the length of the wire L is given by L = 2(l + b).

💡 WORKED EXAMPLE: Given that the cost per unit length is ₹ 5 and the perimeter of the coil is 64 cm, we need to find the dimensions that will minimize the cost. Let's assume the length of the rectangle is 'l' and the breadth is 'b'. Since the perimeter is 64 cm, we have 2(l + b) = 64, which gives l + b = 32. Also, we know that L = 2(l + b) = 2 * 32 = 64. The cost C is given by C = 5L = 5 * 64 = ₹ 320. To minimize the cost, we need to minimize the length of the wire, which is a square of side 16 cm.

⚠️ COMMON MISTAKE: Students often get confused between the perimeter and the length of the wire, which can lead to incorrect calculations. Another common mistake is not using the constraint to express one variable in terms of the other, which can make the problem unsolvable.