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Answer to Question #20742 in Abstract Algebra for daksh

Question #20742
semi circulat region and square region have equal perimeter area of a square exceeds area of semicircular by 7 cm find the perimeter and area
Expert's answer
Let a be side of square. Then square perimeter is 4a and square area is a^2Let r be radius of semicircle. Semicircular perimeter then is pi*r and semicircular area ispi*r^2 / 2
So we have4a=pi*r+2r = r(pi+2) (1)
a^2 = pi*r^2 / 2 + 7 (2)

from (1) we find
a = (pi+2)*r/4
substituting to (2) gives
(pi+2)^2 r^2 / 16 = pi* r^2 / 2 + 7
(pi+2)^2 / 8 * r^2 = pi * r^2 + 14
((pi^2+4pi+4)/8 - pi) r^2 = 14
0.1629 r^2 = 14
r = 9.27
hence
perimeter of semicircle and square is 47.66
area of square is 141.97
are of circle is 134.97

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