2620.sagews

# A mező sugara r
# A kecske kötelének hossza k*r
# A kecske által lelegelt terület és a mező területének aránya q

C1 = circle((0,0), 5, edgecolor = 'green')
C1.set_aspect_ratio(1)

C2 = circle((5,0), 3, edgecolor = 'red')
C2.set_aspect_ratio(1)

O1 = text('O1', (-0.3,-0.3), rgbcolor = 'green')
O2 = text('O2', (5.3,-0.3), rgbcolor = 'red')
A = text('A', (4.25,3.25), rgbcolor = 'blue')
B = text('B', (4.25,-3.25), rgbcolor = 'blue')

l1 = line([(0,0), (4.1,0.3*sqrt(91))], rgbcolor = 'blue')
l2 = line([(0,0), (4.1,-0.3*sqrt(91))], rgbcolor = 'blue')
l3 = line([(5,0), (4.1,0.3*sqrt(91))], rgbcolor = 'blue')
l4 = line([(5,0), (4.1,-0.3*sqrt(91))], rgbcolor = 'blue')
l5 = line([(4.1,0.3*sqrt(91)), (4.1,-0.3*sqrt(91))], rgbcolor = 'blue')

plot(C1 + C2 + O1 + O2 + A + B + l1 + l2 + l3 + l4 + l5)

var("k,q,r,x,y")

(k, q, r, x, y)

solve([x^2 + y^2 == r^2,(x-r)^2 + y^2 == (k*r)^2], x, y)

[[x == -1/2*(k^2 - 2)*r, y == -1/2*sqrt(-k^2 + 4)*k*r], [x == -1/2*(k^2 - 2)*r, y == 1/2*sqrt(-k^2 + 4)*k*r]]

x = -1/2*(k^2 - 2)*r
y = 1/2*sqrt(-k^2 + 4)*k*r

a1 = 2*y
m1 = x
T_haromszog1 = a1*m1/2

alpha = 2*arccos(x/r)
T_korcikk1 = r^2*pi*alpha/(2*pi)

T1 = T_korcikk1 - T_haromszog1

a2 = 2*y
m2 = r - x
T_haromszog2 = a2*m2/2

beta = 2*arccos((1 - x/r)/k)
T_korcikk2 = (k*r)^2*pi*beta/(2*pi)

T2 = T_korcikk2 - T_haromszog2

expand((T1 + T2)/(r^2*pi))

k^2*arccos(1/2*k)/pi - 1/2*sqrt(-k^2 + 4)*k/pi + arccos(-1/2*k^2 + 1)/pi

plot(expand((T1 + T2)/(r^2*pi)), 0, 2)

solve(k^2*arccos(1/2*k)/pi - 1/2*sqrt(-k^2 + 4)*k/pi + arccos(-1/2*k^2 + 1)/pi == q, k)

[k == 1/4*(sqrt(-k^2 + 4) - sqrt(16*pi*q*arccos(1/2*k) - 16*pi*arccos(1/2*k) - k^2 + 16*arccos(1/2*k^2 - 1)*arccos(1/2*k) + 4))/arccos(1/2*k), k == 1/4*(sqrt(-k^2 + 4) + sqrt(16*pi*q*arccos(1/2*k) - 16*pi*arccos(1/2*k) - k^2 + 16*arccos(1/2*k^2 - 1)*arccos(1/2*k) + 4))/arccos(1/2*k)]