Água

Areia

\mbox{tempo} \scriptsize \mbox{ (do percurso ABC)}

abcissa de C

Animação

\small\bold\textcolor{Red}{R_{s}}

abcissa de C

\small\bold\textcolor{OliveGreen}{R_{v}}

\textcolor{Red}{\mbox{Quem tem conhecimento das funções trigonométricas pode notar que }R_{s}=\frac{\mbox{Sen }\beta}{\mbox{Sen }\alpha}}

\textbf{Qual o percurso mais rápido para o nadador salvador chegar ao banhista?} \ \small \mbox{Experimente mover o ponto C. A e B são igualmente manipuláveis}.

Avançar

Recuar

\textbf{Procure localizar o C que torna o percurso B-C-A o mais rápido possível}. \
\small \mbox{Para esse C compare }R_{s} \mbox{ e } R_{v}.

J = (130, -25)
J_2 = (200, 83)
I_2 = (-37, -59)
S_1: (x(I_2) - 100, 0.5 * y(I_2) + 0.5 * y(J_2))
T_1: (x(I_2) * 0.65 + x(J_2) * 0.35, y(J_2) * 0.5 + y(I_2) * 0.5)
U_1: (x(I_2) * 0.65 + x(J_2) * 0.35, y(I_2) - 100)
W_1: (x(I_2) * 0.65 + x(J_2) * 0.35, y(J_2) + 100)
Z_1: (x(I_2) * 0.65 + x(J_2) * 0.35, y(J_2) * 0.5 + y(I_2) * 0.5)
A_2: (x(I_2) - 100, y(I_2) * 0.5 + y(J_2) * 0.5)
V_1: (x(I_2) - 100, y(J_2) + 100)
polígono2, v_1, w_1, z_1, a_2 = Polygon(V_1, W_1, Z_1, A_2)
A = PointIn(polígono2)
O: (0, y(I_2) * 0.5 + y(J_2) * 0.5)
R_1: (x(I_2) - 100, y(I_2) - 100)
polígono1, r_1, s_1, t_1, u_1 = Polygon(R_1, S_1, T_1, U_1)
B = PointIn(polígono1)
p = Function((x(A) - x) / sqrt((x - x(A))^(2) + (y(A) - y(O))^(2)) / ((x - x(B)) / sqrt((x - x(B))^(2) + y(B - y(O))^(2))), Min[x(B), x(A)], Max[x(B), x(A)])
q = Function(y(J) + 5 * p(x - x(J)), Min[x(B), x(A)] + x(J), Max[x(B), x(A)] + x(J))
k = Segment((x(B), y(I_2) * 0.5 + y(J_2) * 0.5), (x(A), y(I_2) * 0.5 + y(J_2) * 0.5))
C = Point(k)
M: (x(C), (x(A) - x(C)) / sqrt((x(C) - x(A))^(2) + y(A)^(2)) / ((x(C) - x(B)) / sqrt((x(C) - x(B))^(2) + y(B)^(2))))
vel_{natação} = 3.9 (slider from 1 to 10, step 0.1)
vel_{corrida} = 7.4 (slider from 1 to 10, step 0.1)
F_2 = (195, -25)
Q = (185, -25)
j = OrthogonalLine(C, a)
I = Point(j)
α = If(x(B) < x(A), Angle[B, C, I], Angle[I, C, B])
H = Point(j)
β = If(x(B) < x(A), Angle[A, C, H], Angle[H, C, A])
rang: sin(β) / sin(α)
G_2: (x(F_2), y(F_2) + 5 * rang)
Gt = If(rang < 100, G_2, (195, 1000))
R: (x(Q), y(Q) + 5 * rang)
Rt = If(rang < 100, R, (185, 1000))
b: x < 30
c: y < 0
dd = Extremum(((x - x(B))^(2) + x(A)^(2))^(0.5) / vel_{natação} + ((x - x(A))^(2) + y(A)^(2))^(0.5) / vel_{corrida}, -100, 100)
S: (x(dd), 0)
d = Segment(A, C)
e = Segment(C, B)
f = Segment(B, A)
distânciaBC = Distance(A, C)
TextoBC: "\overline{" + (Name[A]) + (Name[C]) + "} \, = \, " + distânciaBC
distânciaCA = Distance(C, B)
TextoCA: "\overline{" + (Name[C]) + (Name[B]) + "} \, = \, " + distânciaCA
tempo: distânciaBC / vel_{corrida} + distânciaCA / vel_{natação}
D_3 = (70, -25)
F: (x(D_3), y(D_3) + 90)
u = Vector(D_3, F)
G: (x(D_3) + 40, y(D_3))
v = Vector(D_3, G)
texto5: "  " + tempo + " s"
bol = true
rvel: vel_{corrida} / vel_{natação}
L: (x(J), y(J) + 90)
K: (x(J) + 40, y(J))
w = Vector(J, K)
N: (x(J) + x(M), q(x(J) + x(M)))
z = Vector(J, L)
O_1 = (215, -25)
P: (x(O_1), y(O_1) + 5 * rvel)
l = Segment(O_1, P)
m = Segment(R, Q)
V: (x(A), y(A))
n = Line(C, A)
Z = Point(n)
s = Circle(Z, 10)
A_1 = Intersect(s, n, 2)
U: (x(B), y(B))
C_1: (x(A), y(A))
D_1: (x(C), y(I_2) * 0.5 + y(J_2) * 0.5 + 10)
E_1: (x(C), y(I_2) * 0.5 + y(J_2) * 0.5 - 10)
t = Segment(D_1, E_1)
F_1: (x(B), 0)
c_1 = Circle(F_1, 5)
G_1 = Point(c_1)
H_1 = Point(c_1)
d_1 = Semicircle(H_1, G_1)
I_1 = Point(d_1)
texto12: "\textcolor{OliveGreen}{" + (LaTeX[rvel]) + "}"
texto11: "\bold\textcolor{Red}{" + (LaTeX[rang]) + "}"
W: (x(V) + 10, y(A) - 5)
B_1: (x(C_1) - 10, y(C_1) - 5)
a_1 = Segment(O, C)
J_1: (x(D_3) + x(C) - x(O), y(D_3))
b_1 = Segment(D_3, J_1)
K_1: (x(D_3) - 6, y(D_3))
e_1 = Segment(D_3, K_1)
L_1: (x(D_3), y(D_3) - 5)
f_1 = Segment(L_1, D_3)
M_1: (x(J) - 5, y(J))
N_1: (x(J), y(J) - 5)
g_1 = Segment(J, M_1)
h_1 = Segment(N_1, J)
P_1: (x(J) + x(C), y(J))
i_1 = Segment(P_1, J)
Q_1 = Point(b)
B_2 = (-200, 0)
C_2: (x(I_2) * 0.65 + x(J_2) * 0.35, y(J_2) * 0.5 + y(I_2) * 0.5)
j_1 = Segment(A_2, C_2)
D_2 = (225, -25)
E_2: (x(D_2), y(D_2) + 5 * rvel)
polígono3, d_2, e_2, p_1, o_1 = Polygon(D_2, E_2, P, O_1)
r = Function(y(J) + 5 * vel_{corrida} / vel_{natação}, x(J) - 10, x(J) + 40)
polígono4, k_1, l_1, m_1, n_1 = Polygon(F_2, Gt, Rt, Q)
g = Function(sqrt((x - x(B))^(2) + (y(B) - y(O))^(2)) / vel_{natação} + sqrt((x - x(A))^(2) + (y(A) - y(O))^(2)) / vel_{corrida}, Min[x(B), x(A)], Max[x(B), x(A)])
Max = Max(g, Min[x(B), x(A)], Max[x(B), x(A)])
Min = Min(g, Min[x(B), x(A)], Max[x(B), x(A)])
max: y(Max)
min: y(Min)
minap: min - 0.1
texto13: "" + minap + ""
maxap: max + 0.1
h = Function(90 / (maxap - minap) * (g(x - 70) - minap) - 25, Min[x(B), x(A)] + 70, Max[x(B), x(A)] + 70)
texto14: "" + maxap + ""
texto6: "\small\bold\textcolor{OliveGreen}{R_{v}=\frac{\mbox{vel corrida}}{\mbox{vel natação}}\\\;\;\;\;\;\;=" + (LaTeX[rvel]) + "}"
H_2: (170, r(170))
q_1 = Segment(N, Rt)
b_2 = Segment(H_2, P)
T: (x(B) - 9, y(B))
i = 0 (slider from 0 to 1, step 1)
K_2 = Corner(5)
D: (x(A), y(I_2) * 0.5 + y(J_2) * 0.5)
E: (x(B), y(I_2) * 0.5 + y(J_2) * 0.5)
E_3: (x(D_3) + x(C) - x(O), h(x(D_3) + x(C) - x(O)))
texto7: "\small\bold\textcolor{Red}{R_{s}=\frac{\;\;\frac{\mbox{ }\overline{DC}\mbox{ }}{\overline{AC}}\;\;}{\frac{\mbox{ }\overline{EC}\mbox{ }}{\overline{BC}}{}}\\\;\;\;\;\;\;=" + (LaTeX[rang]) + "}"