Question

consider a wire 2 ft long cut into two pieces. one piece forms a circle with radius r and the other forms a square with side length x. (a) determine a formula for the radius r in terms of x .

Answer 1

The formula for the radius r in terms of x . and for the maximum areas is x=2/$\pi$+4

Given that,

y forms a circle of radius r

y=2$\pi$r

r=y/2$\pi$

(2-y)- forms Square Side x

(2-y) = 4x

x=(2-y)/4

Now Sum of Area's=Area of Square +Area of Circle

Sum = $\pi$r² + x²

Substitute the r and x values in above equation,

A(y)= y²/4$\pi$+(y-2)²/ 16

To maximize Area A(y)

A'(y)= 0

2y/4$\pi$ + 2(y-2)/16 =0

y/2$\pi$ + (y-2)/8 =0

y = 2$\pi$/$\pi$+4

Y max will be max, x to be maximum.

for maximum sum of areas,

x=2/$\pi$+4

Hence,The formula for the radius r in terms of x . and for the maximum areas is x=2/$\pi$+4

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