* If length of the curve
BEC under blue shed is
a1, then a1=2πb(360-A)/360 where A is in
degree.
=(2π/360)*b*(360-A)=k*b*(360-A) ...(1) and k is a
constant. k=(2π/360)
a1=(2π-A)*b .....(1a)
where A is in radian.
=f1(A)*f2(b)
* If length of the curve BDC under white shed is a2,
then a2= (2πb-a1) =2πb(A)/360 =k*b*A
....(2) where A is in
degree.
a2= (2πb-a1)= A*b ......(2a)
where A is in radian.
* Length of the chord BC is a and
a=√2*b *√(1-cosA)
* a1/a =(√2π /360)*(360-A)/ √(1-cosA)
where A is in degree. a1/a =(2π-A)/[√2* √(1-cosA)]
where A is in radian. * a2/a =(√2π /360)*A/ √(1-cosA)
where A is in degree. a2/a =A/[√2* √(1-cosA)]
where A is in radian. *(a1+a2)/a = √2π / √(1-cosA)
* Area of the curved triangle is b^{2}*ar/2
and area of straight triangle is BC*height/2 where BC=2bsin(A/2)
and height is bcos(A/2). Hence area is A= (b^{2}sin(ar))/2.
Therefore, Area Difference =Δarea=Curved Area-Straight area =
(b^{2}/2)*angle A* (1-[sinA/A]) .
If we take function f(b) = (b^{2}/2) and
function f''(A) = A* (1-[sinA/A]), then
Area Difference =ΔA= f(b)*f'(A).
First function is dependent on radius and second one dependent on
angle A and product of both the functions contribute to area
difference between the curved and the straight triangle. When A is
very small close to zero radian, f(A) -->
zero as sinA/A --> 1 and therefore Area Difference =Δarea
--> zero which implies that the curved arc almost coincides with the
straight chord. On the other hand, Δarea / area of curve BCD= = 1-(sinA/A) which implies that the ratio is
only dependent on Angle and not on radius as the effect of change
due to radius are equal for both numerator and denominator and
cancel out. On limiting value of A--> zero, this ratio tends to
zero.
* On filling up values of x1,x2 and submitting, we get 6 chords
[C(4,2)] connecting points , B(x1,y1a) ,B1(x1,y1b),
C(x2,y2a) & C1(x2,y2b). |