A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?

Suppose the given wire, which is to be made into a square and a circle, is cut into two pieces of length $x$ and $y$ $m$ respectively. Then,

\[x\text{ }+\text{ }y\text{ }=\text{ }28\text{ }\Rightarrow \text{ }y\text{ }=\text{ }\left( 28\text{ }-\text{ }x \right)\]

We know that perimeter of square, \[4\text{ }\left( side \right)\text{ }=\text{ }x\]

\[Side\text{ }=\text{ }x/4\]

Area of square \[=\text{ }{{\left( x/4 \right)}^{2}}~=\text{ }{{x}^{2}}/16\]

Circumference of circle, \[2\text{ }\pi \text{ }r\text{ }=\text{ }y\]

\[r\text{ }=\text{ }y/\text{ }2\text{ }\pi \]

RD Sharma Solutions for Class 12 Maths Chapter 18 Maxima and Minima Image 26

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