Q. 333.7( 3 Votes )

# A wire of length 36 cm is cut into two pieces. One of the pieces is 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 be minimum.

Answer :

Given,

• Length of the wire is 36 cm.

• The wire is cut into 2 pieces.

• One piece is made to a square.

• Another piece made into a equilateral triangle.

Let us consider,

• The perimeter of the square is x.

• The perimeter of the equilateral triangle is (36-x).

• Side of the square is

• Side of the triangle is

Let the Sum of the Area of the square and triangle is

--- (1)

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. __This is because if the function A(x) has a maximum/minimum at a point c then A’(c) = 0.__

Differentiating the equation (1) with respect to x:

[Since ]

----- (2)

To find the critical point, we need to equate equation (2) to zero.

__Now to check if this critical point will determine the minimum area, we need to check with second differential which needs to be positive.__

Consider differentiating the equation (3) with x:

----- (4)

[Since ]

Now let us find the value of

As , so the function A is minimum at

Now, the length of each piece is and

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