Respuesta :
The triangle should have 4.35 metres of wire, which means the square should have 6.978 meters
Let "x" is the cutting distance from one end of the wire.
Then, area of square would be ,
Area = (x/4)²
The side of equilateral triangle would be,
L=(10-x)/3
height would be (10-x)/3*√(3)/2
=> (10-x)√(3)/6
So, the area of triangle is
Area of triangle= ((10-x)/3)² ×√(3)/4
=> (10-x)² ×√(3)/36
So, the total area is ,
A = (x/4) + (10 - x)²/36√3
Differentiating Area w.r.t x
dA/dx=x/8-2 ×√(3)/36(10-x)
=> x/8+20× √(3)/36-2 × √(3)/36x
Putting the expression equal to 0,
x/8+20× √(3)/36-2 × √(3)/36x = 0
=> x(1/8+ √(3)/18
=> 10√(3)/18
x*0.22122 = 0.96225
x=4.35
Substituting the value of x
This corresponds to minimum area which is:
A(4.35)=(4.35/4)^2+(10-4.35)^2 × √(3)/36=1.183+1.536
=> 2.719m²
Thus , the minimum area is 2.71 square meters when x=4.35 m. As a result, the wire should be cut such that the dimension of the square is formed by the piece having a length of 4.35 m, and the equilateral triangle is formed by the piece having a length 5.65 m.
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