Problem: A foreman noticed an inspector checking a 3 β² β² 3 β² β² 2 β² β² 2 β² β² 1 β² β² 1 β² β² d d
Answer Choices:
A. 0.87 0 . 8 7
B. 0.86 0 . 8 6
C. 0.83 0 . 8 3
D. 0.75 0 . 7 5
E. 0.71 0 . 7 1 Solution:
O A βΎ = 1 2 + r , O β² A βΎ = 1 + r , B A βΎ = 1 1 2 β r O A = 2 1 β + r , O β² A = 1 + r , B A = 1 2 1 β β r
Area ( β³ O B A ) = 2 ( β³ O B A ) = 2 ( β³ B O β² A ) ( β³ B O β² A )
Semiperimeter (\triangle \mathrm{OBA})=\dfrac{1}{2}\left(\dfrac{1}{2}+r+1+\dfrac{3}{2}-r\right)=\dfrac{3}
Semiperimeter \left(\triangle B^{\prime} O A\right)=\dfrac{1}{2}\left(\dfrac{3}{2}-r+\dfrac{1}{2}+1+r\right)=\dfrac{3}
β΄ 3 2 ( 1 β r ) ( 1 2 ) r = 2 3 2 ( r ) ( 1 ) ( 1 2 β r ) β΄ 7 r = 3 , r = 3 7 , d = 8 7 β . 86 β΄ 2 3 β ( 1 β r ) ( 2 1 β ) r β = 2 2 3 β ( r ) ( 1 ) ( 2 1 β β r ) β β΄ 7 r = 3 , r = 7 3 β , d = 7 8 β β . 8 6
or
Draw the altitude h h A A
\left(\dfrac{1}{2}+r\right)^{2}-t^{2}=\left(\dfrac{1}{2}-r\right)^{2}-(1-t)^{2}, t=2 r-\dfrac{1}{2}, h^{2}=3 r-3 r^
Since Area ( Ξ O A O β² ) = 1 2 h β
3 2 = ( 3 2 + r ) ( r ) ( 1 ) ( 1 2 ) ( Ξ O A O β² ) = 2 1 β h β
2 3 β = ( 2 3 β + r ) ( r ) ( 1 ) ( 2 1 β ) β 9 16 h 2 = 9 16 ( 3 r β 3 r 2 ) = 3 4 r + 1 2 r 2 β΄ r = 3 7 1 6 9 β h 2 = 1 6 9 β ( 3 r β 3 r 2 ) = 4 3 β r + 2 1 β r 2 β΄ r = 7 3 β
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