Respuesta :
Answer:
A(t) = 41,47 in²
Step-by-step explanation:
Let´s call "x" the cut of point to get to pieces of wire, we make a square from x and the regular octagon will be shaped with 24-x
Then Area of the square A(s) = x²
Area of the octagon is A(o) = 1/2*p*length of apothem (d)
p = ( 24 - x )
length of apothem (d) :
The side of the octagon is equal to ( 24 - x ) / 8 half the side is
( 24 - x ) / 16
tan α = ( 24- x ) 16 / d since ∡s in octagon are 360 / 8 = 45°
α ( ∡ between apothem and one of the interiors ∡ of the octagon )half of 45 is α = 22,5°
tanα = 0,41
d = (24 - x ) / 16*0,41 d = ( 24 - x ) / 6,56
Then
A(t) = A(s) + A(o)
A(t) = x² + (1/2)* ( 24 - x ) ( 24 - x ) / 6,56
Note A(t) = A(x)
A(x) = x² + (1/2) * (24 - x )²/ 6,56
A(x) = x² + ( 1/ 2*6,56) * ( (24)² -48*x + x² )
Taking derivatives on both sides of the equation
A´(x) = 2*x + ( 1/13,12)* ( - 48 + 2x )
A´(x) = 2*x - 48/ 13,12 + 2*x
A´(x) = 4*x - 3,66
A´(x) = 0 4x = 3,66 x = 0,91 in and d =( 24 - x ) / 6,56
d = ( 24 - 0,91 ) / 6,56 d = 3,52
Then A(s) = (0,91)² A(s) = 0,83 in²
A(o) = 1/2 * ( 24 - 0,91 )* 3,52
A(o) = 40,63 in²
A(t) = 40,63 + 0,83
A(t) = 41,47 in²
