GATE EC 2016 Set 2 — Question 10
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
A 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?
Correct answer
(B) 40
Solution
Let the length of the wire be mm. Suppose the wire is cut into two pieces of lengths and .
1.Square: The piece of length is used to form a square. The perimeter is , so the side . The area of the square is .
2.Rectangle: The piece of length is used to form a rectangle with sides in the ratio . Let the sides be and . The perimeter is , so . The area of the rectangle is .
3.Total Area: The combined area is .
4.Minimization: To find the minimum area, we differentiate with respect to and set it to zero:
5.Side of the Square: The length of the side of the square is mm.
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