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 len

Question

2 years ago

11

gth x. (a) determine a formula for the radius r in terms of x .

1 answer:

makkiz [27] · Answer 1 · 2023-12-20T09:21:04+03:00

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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