A Wire is 100cm long is cut into two pieces, one of which is bent into a circle and the other into a shape of square.?

Let x cm be the length of the piece of wire bent into a circle.
Hence, the length of the piece of wire bent into a square = (100 - x) cm

Circumference of the circle in cm:
x = 2πr
Then, r = x/2π

Area of the circle in cm²
= πr²
= π(x/2π)²
= x²/(4π)

Area of the square in cm²
= [(100 - x)/4]²
= (x² - 200x + 10000)/16

Total enclosed area in cm², A = [x²/(4π)] + [(x² - 200x + 10000)/16]
dA/dx = [x/(2π)] + [(x - 100)/8)]
d²A/dx² = [1/(2π)] + (1/8)

When dA/dx = 0:
[x/(2π)] + [(x - 100)/8)] = 0
[4x/(8π)] + [xπ - 100π)/(8π)] = 0
4x + xπ - 100π = 0
x = 100π/(4 + π)
100 - x = 100 - [100π/(4 + π)]
100 - x = 400/(4 + π)

When x = 100π/(4 + π):
dA/dx = 0
d²A/dx² = [1/(2π)] + (1/8) > 0
Hence, minimum enclosed area when x = 100π/(4 + π)

When the sum of the enclosed area is the minimum:
The length of the piece of wire bent into a circle = 100π/(4 + π) cm ≈ 43.99 cm
The length of the piece of wire bent into a square = 400/(4 + π) cm ≈ 56.01 cm