Answer:
Used wire in circle x = 2.64 m
Used in square L - x = 3.36 m
Total wire used 6 m
Step-by-step explanation:
We have a wire of 6 meters long.
We will cut it a distance x from one end, to get two pieces
x and 6 - x
We are going to use the piece x to get the circle then
So Perimetr of a circle is 2π*r (r is the radius of the circle) then:
x = 2*π*r ⇒ r = x/2*π
And area would be A(c) = π* (x/2*π)² ⇒ A(c) = x²/4π
From 6 - x we will get a square, and as the perimeter is 4 times the side
we have
( 6 - x )/ 4 is the side of the square
And the area is A(s) = [( 6 - x ) /4]²
Total area as function of x is
A(t) = A(c) + A(s)
A(x) = x²/4π + [ ( 6 - x ) / 4 ]²
A(x) = x²/4π + (36 + x² - 12x) /16
A(x) = 1 / 16π [ 4x² + 36π + πx² - 12π x ]
Taking drivatives on both sides of the equation we get:
A´(x) = 1/ 16π [8x +2πx - 12π]
A´(x) = 0 ⇒ 1/ 16π [8x +2πx - 12π] = 0
[8x +2πx - 12π] = 0
8x + 6.28x - 37.68 = 0
14.28x - 37.68 = 0 ⇒ x = 37.68 /14.28
x = 2.64 m length of wire used in the circle
Then the length L for the side of the square is
(6 - x )/4 ⇒ ( 6 - 2.64 )/ 4 ⇒ 3.36 / 4
L = 0.84 m total length of wire used in the square is
3.36 m
And total length of wire used is 6 m
The function is a quadratic function and "a" coefficient is positive then is open upward parabola there is not a maximun
