Project ForgeA wire of length 340 mm is to be cut into two parts. One of the parts is to be made into a square and the other into a rectangle where sides are in the ratio of 1:2. What is the length of the side of the square (in mm) such that the combined area of the square and the rectangle is a MINIMUM?
- A30
- 40
- C120
- D180
Explanation
Let the square's side be s and the rectangle's shorter side be x (so its sides are x and 2x). Wire constraint: 4s + 6x = 340, so s = 85 - 1.5x. Combined area A(x) = s^2 + 2x^2 = (85-1.5x)^2 + 2x^2. dA/dx = -3(85-1.5x) + 4x = -255 + 4.5x + 4x = 8.5x - 255 = 0, giving x = 30. Then s = 85 - 1.5(30) = 40. The second derivative is positive (sum of squares is convex), confirming this is a minimum.
Source: GATE ME 2016 Set-3 (2016)
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