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ABC is a variable isosceles triangle with AB=AC touching a semicircle of radius a cm at P and Q. O is the centre of the semicircle and BOC is a straight line. Let S cm^2 be the area of △ABC and ∠BAO=θ.

a) Show that
S=2a^2/sin2θ   ,where 0<θ<π/2
b) Determine the range of values ofθ for which S is
i) increasing,and
ii)decreasing.
Hence, find the minimum values of S

c) Sketch the curve of S againstθ for 0<θ<π/2.

d) If 2a<AO<3a , find the greatest value of S

a )計到,,b)c)d)唔識

Naozumi

(b) d/dθ  (1/sin 2θ ) = -(2cos 2θ )/(sin 2θ )^2
所以只需要睇 cos 2θ 個sign就 d/dθ 係 >0 定 < 0

明顯,當0<θ <π/4,  cos 2θ  > 0,  即 d/dθ  (1/sin 2θ ) < 0
當π/4<θ <π/2,  cos 2θ  < 0,  即 d/dθ  (1/sin 2θ ) > 0

所以個min係θ = π/4

(d) 只係限制左個θ個range,
2a<AO<3a
2a<a/sinθ <3a
1/3<sinθ<1/2
arcsin 1/3 <θ< π/6
由(b), 當 arcsin 1/3 <θ< π/6, S係decreasing,所以max S occur at θ = sin 1/3

紅色部份: 昨日計錯了,現已修改

[ 本帖最後由 Naozumi 於 2009-6-3 11:14 AM 編輯 ]