Answer :
The radius in terms of x is 2/(π+4)
What is area?
Area, a quantity that describes the extent of an area on a plane or curved surface. The area of a planar region refers to the area of a shape or planar layer, while surface area refers to the area of an open surface or boundary of a three-dimensional object.
Given that,
y forms a circle of radius r
y = 2πr
r = y/2π
(2-y) forms Square Side x
(2-y) = 4x
x = (2-y)/4
Now Sum of Areas = Area of Square + Area of Circle
Sum = πr² + x²
Substitute the values of r and x in above equation:
A(y) = y²/4π + (y-2)²/ 16
To maximize Area A(y)
A'(y) = 0
2y/4π + 2(y-2)/16 = 0
y/2π + (y-2)/8 =0
y = 2π/(π+4)
For maximum sum of areas,
x = 2/π + 4
Hence, The formula for the radius r in terms of x and for the maximum areas is x=2/(π+4)
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