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Question
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
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Solution
\[\text {Suppose the wire, which is to be made into a square and a triangle, is cut into two pieces of length x and y, respectively . Then}, \]
\[x + y = 20 .......... \left( 1 \right)\]
\[\text { Perimeter of square }, 4\left( Side \right) = x\]
\[ \Rightarrow \text { Side } = \frac{x}{4}\]
\[\text { Area of square = }\left( \frac{x}{4} \right)^2 = \frac{x^2}{16}\]
\[\text { Perimeter of triangle }, 3\left( \text { Side } \right) = y\]
\[ \Rightarrow \text { Side } = \frac{y}{3}\]
\[\text { Area of triangle } = \frac{\sqrt{3}}{4} \times \left( \text { Side } \right)^2 = \frac{\sqrt{3}}{4} \times \left( \frac{y}{3} \right)^2 = \frac{\sqrt{3} y^2}{36}\]
\[\text { Now,} \]
\[z =\text { Area of square + Area of triangle }\]
\[ \Rightarrow z = \frac{x^2}{16} + \frac{\sqrt{3} y^2}{36}\]
\[ \Rightarrow z = \frac{x^2}{16} + \frac{\sqrt{3} \left( 20 - x \right)^2}{36} .................\left[ \text { From eq } . \left( 1 \right) \right]\]
\[ \Rightarrow \frac{dz}{dx} = \frac{2x}{16} - \frac{2\sqrt{3}\left( 20 - x \right)}{36}\]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow \frac{2x}{16} - \frac{\sqrt{3}\left( 20 - x \right)}{18} = 0\]
\[ \Rightarrow \frac{9x}{4} = \sqrt{3}\left( 20 - x \right)\]
\[ \Rightarrow \frac{9x}{4} + x\sqrt{3} = 20\sqrt{3}\]
\[ \Rightarrow x\left( \frac{9}{4} + \sqrt{3} \right) = 20\sqrt{3}\]
\[ \Rightarrow x = \frac{20\sqrt{3}}{\left( \frac{9}{4} + \sqrt{3} \right)}\]
\[ \Rightarrow x = \frac{80\sqrt{3}}{\left( 9 + 4\sqrt{3} \right)}\]
\[ \Rightarrow y = 20 - \frac{80\sqrt{3}}{9 + 4\sqrt{3}} .............\left[ \text { From eq }. \left( 1 \right) \right]\]
\[ \Rightarrow y = \frac{180}{9 + 4\sqrt{3}}\]
\[ \frac{d^2 z}{d x^2} = \frac{1}{8} + \frac{\sqrt{3}}{18} > 0\]
\[\text { Thus, z is minimum when } x = \frac{80\sqrt{3}}{\left( 9 + 4\sqrt{3} \right)} \text { and }y = \frac{180}{9 + 4\sqrt{3}} . \]
\[\text { Hence, the wire of length 20 cm should be cut into two pieces of lengths }\frac{80\sqrt{3}}{\left( 9 + 4\sqrt{3} \right)} \text { m and } \frac{180}{9 + 4\sqrt{3}} m . \]
Notes
The solution given in the book is incorrrect . The solution here is created according to the question given in the book.
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