 Hi, and welcome to the session. Let us discuss the following question. The question says, a wire of length 36 centimeters is cut into two pieces, one of the pieces so turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two-winged minima. So according to this question, a wire of length 36 centimeters is cut into two pieces, one of the pieces is turned in the form of a square and the other piece is turned in the form of an equilateral triangle. We have to find the length of each piece such that sum of areas of both these pieces and the minima. So now let x centimeters be the length of a side of the square and y centimeters, the length of side of an equilateral triangle. Into question, the square, that is 4x, plus perimeter of the equilateral triangle, that is 3y is equal to 36, right? This implies y is equal to 36 minus 4x by the equilateral triangle squared. So a is equal to squared plus 3 by 4 by squared, substituting value of y squared plus 3 by 4 into 36 minus 4x by 3 whole squared. Equation as equation number 1, differentiating both sides of 1 with respect to x, dA by dx equals to 2x plus 3 by 4 into 2 into 36 minus 4x into minus 4 by 9. This implies dA by dx is equal to 2x minus 2 root 3 by 9 into 36 minus 4x. Now for maximum or minimum, dA by dx equals to 0. This implies 2x minus 2 root 3 by 9 into 36 minus 4x is equal to 0. This implies 2x is equal to 2 root 3 by 9 into 36 minus 4x. Now this implies 2x is equal to 8 root 3 minus 8 root 3x by 9. This implies 8 root 3 by 9 into x is equal to 8 root 3. This implies 18 plus 8 root 3 by 9 into x is equal to 8 root 3. This implies x is equal to 8 root 3 into 9 by 18 plus 8 root 3. Now this is equal to 8 root 3 into 9 by 2 into 9 plus 4 root 3. This is equal to 36 root 3 by 9 plus 4 root 3. So x is equal to 36 root 3 by 9 plus 4 root 3. This equation has equation number 2. Differentiating respect to x, we get d2A by dx2 equals to minus 2 root 3 by 9 into 0 minus 4. Now this is equal to 2 plus 8 root 3 by 9 is greater than 0. We can see that value of d2A by dx2 equals to 36 root 3 by 9 plus 4 root 3 is positive. Hence by second derivative test, minimum 36 root 3 by 9 plus 4 root 3. We know that y is equal to 36 minus 4x by 3. So by substituting value of x, we get y as 36 minus 4 into 36 root 3 by 9 plus 4 root 3 divided by 3. This is equal to 324 plus 144 root 3 minus 144 root 3 by 3 into 9 plus 4 root 3. Now this is equal to 324 by 3 into 9 plus 4 root 3. And this is equal to 108 by 9 plus 4 root 3. We have to find the length of each piece. The length of square is equal to perimeter of square. The length of square is 4x66 into root 3 by 9 plus 4 root 3. This is equal to 144 root 3 by 9 plus 4 root 3. The length of square is 144 root 3 by 9 plus 4 root 3 centimeter. Now we will find length of equilateral triangle into perimeter of equilateral triangle is equal to 3y. Now y is equal to 108 by 9 plus 4 root 3 and this is equal to 324 by 9 plus 4 root 3. So length of equilateral triangle is 324 by 9 plus 4 root 3. Hence I required answers are 144 root 3 by 9 plus 4 root 3 centimeters and 324 by 9 plus 4 root 3 centimeters. So this completes the session. Bye and take care.