GATE ME 2016 Set 3 — Question 10
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Engineering Mathematics → Quantitative Aptitude → 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 total length of the wire be mm.
Let the wire be cut into two parts of lengths and .
Perimeter of square =
Side of square,
Area of square,
Let the sides be and .
Perimeter of rectangle =
Side
Area of rectangle,
To find the minimum area, differentiate with respect to and set it to zero:
Substituting mm:
Thus, the length of the side of the square for minimum combined area is 40 mm.
Let the wire be cut into two parts of lengths and .
1.Square part:
Let the length be used to form a square.Perimeter of square =
Side of square,
Area of square,
2.Rectangle part:
Let the length be used to form a rectangle with sides in the ratio .Let the sides be and .
Perimeter of rectangle =
Side
Area of rectangle,
3.Total Area Optimization:
Total Area To find the minimum area, differentiate with respect to and set it to zero:
4.Side of the square:
The side of the square is Substituting mm:
Thus, the length of the side of the square for minimum combined area is 40 mm.
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