Answer:
s = (5 - πr)/2
r = 5/(4(1+π))
r is 5/(4π) to obtain minimal area
Step-by-step explanation:
Given
Length of wire = 10cm
The length of the wire is being divided between the perimeter of the square and the circumference of the circle.
Let s = length of a side of the square.
Let r = Radius of the circle
Perimeter of Square = 4s
Circumference b = 2πr
Perimeter of the Square and circle
4s + 2πr = 10 --- make s the subject of formula
4s = 10 - 2πr
s = (10 - 2πr)/4
s = (5 - πr)/2
Area of the square and circle
πr² + s² = A --- substitute (5-πr)/2 for s
πr² + (5-πr)²/2² = A
πr² + (25 - 10πr + (πr)² )/4= A
A = πr² + (πr)²/4 - 10πr/4 + 25/4
Differentiating with respect to r
dA/dr = 2πr + 2π²r/4 - 10π/4
0 = 2πr + 2π²r/4 - 10π/4
Solve for r
2πr + 2π²r = 10π/4
r(2π + 2π²) = 10π/4
r = 10π/4 / (2π + 2π²)
r = 10π/4 / 2π((1 + π))
r = 5/(4(1+π))
Since A(r) is concave up, it has a minimum at
r = 5/(4(1+π))
s = (5 - πr)/2 and s>0
Since s = (15 - 2πr ) / 4 and s > 0,
the maximum r is 5/(4π).
This same r will also maximize the area.
