Answer:
Piece bent into a triangle = 0.6232 meters,
Piece bent into a circle = 0.3768 meters.
Step-by-step explanation:
The first piece of wire measures X, and the second one measures (1-X).
the first piece is used to make an equilateral triangle. The area of this triangle is A1 = L^2*sqrt(3)/4, where L is the lenght of the sides.
The wire measures X, and we need to make 3 equal sides, so each side will measure X/3, and the area will be A1 = (X/3)^2 * sqrt(3)/4 = X^2 * sqrt(3)/36 = 0.04811*X^2
The second wire is used to make a circle. The area of a circle is A2 = pi*r^2, where r is the radius of the circle.
For this circle, its circunference will have the length of (1-X), and the formula for the length is 2*pi*r, so:
(1-X) = 2*pi*r
r = (1-X)/(2*pi)
The area is pi*r^2, so
A2 = pi*(1-X)^2/(4*pi^2) = (1-X)^2/(4*pi) = 0.07958*(1-X)^2
If we sum A1 and A2, we have the total area A that we want to minimize:
A = A1 + A2 = 0.04811*X^2 + 0.07958*(1-X)^2
A = 0.04811*X^2 + 0.07958*(1-2*X+X^2)
A = 0.12769*X^2 - 0.15916*X + 0.07958
The value of X that gives us the minimum value of A is calculated using the formula:
Xv = -b/2a, where a and b are the coefficients of the quadratic equation.
In our case: a = 0.12769 and b = -0.15916
So, we have that:
Xv = -b/2a = 0.15916/
0.25538 = 0.6232
So the length of the piece bent into a triangle is 0.6232 meters, and the length of the piece bent into a circle is 1-X = 0.3768 meters
