OPTIMISATION
Author: Javet Jean-Philippe
Date: 19.3.2005

 _{200}

 _{100}

 _{300}

 _{400}

 _{500}

 _{600}

x

18 – 2x

x

x

x

22 – 2x

La dérivée s'annule lorsque

le volume est maximal.

V(x)

côté x

D = (-6.5, -7.5)
C = (-2, -7.5)
F = Midpoint(D, C)
M_4 = (-6.49, -0.53)
L_4 = (-0.06666666666666667, -7.5)
U_1 = Point(xAxis)
L_1 = Intersect(xAxis, yAxis)
u_1 = Vector(L_1, U_1)
n_5 = OrthogonalLine(M_4, u_1)
V_1 = Point(yAxis)
v_1 = Vector(L_1, V_1)
m_5 = OrthogonalLine(L_4, v_1)
N_4 = Intersect(n_5, m_5)
p_5 = Segment(F, N_4)
f(x) = x (18 - 2 x) (22 - 2 x) / 80
f_1 = Derivative(f)
B = (-2, -2)
A = (-6.5, -2)
c = Segment(D, F)
g = OrthogonalLine(D, c)
i = OrthogonalLine(B, g)
j = OrthogonalLine(B, i)
H = Midpoint(A, C)
G = Point(p_5)
J = Mirror(G, H)
k = Circle(B, J)
K, L = Intersect(k, j)
n = OrthogonalLine(K, j)
m = OrthogonalLine(J, i)
Q = Intersect(n, m)
a_1 = Segment(Q, J)
u: 4 a_1
f2: f_1(u)
long: 22 - 2 u
larg: 18 - 2 u
a = Segment(A, B)
b = Segment(B, C)
d = Segment(D, A)
E = Midpoint(C, D)
e = Segment(D, E)
I = Midpoint(A, C)
h = Circle(D, G)
M, N = Intersect(h, g)
l = OrthogonalLine(C, j)
p = OrthogonalLine(G, l)
q = OrthogonalLine(N, g)
O = Intersect(n, p)
P = Intersect(q, p)
R = Intersect(q, m)
S = Intersect(l, m)
T = Intersect(p, i)
U = Intersect(g, n)
V = Intersect(j, q)
r = Segment(U, O)
s = Segment(O, Q)
t = Segment(Q, K)
b_1 = Segment(J, T)
c_1 = Segment(T, O)
d_1 = Segment(O, P)
e_1 = Segment(P, R)
g_1 = Segment(R, V)
h_1 = Segment(V, K)
i_1 = Segment(N, P)
j_1 = Segment(N, U)
k_1 = Segment(P, G)
l_1 = Segment(G, S)
m_1 = Segment(S, R)
n_1 = OrthogonalLine(I, xAxis)
W = Point(n_1)
p_1 = OrthogonalLine(W, n_1)
Y = Intersect(p_1, p)
Z = Intersect(g, p_1)
q_1 = Circle(Y, Z)
A_1, B_1 = Intersect(q_1, p)
C_1 = (-8.266666666666666, 4.966666666666667)
r_1 = Segment(W, C_1)
s_1 = OrthogonalLine(A_1, n_1)
D_1 = Intersect(s_1, n_1)
E_1 = Intersect(s_1, m)
F_1 = Intersect(p_1, m)
t_1 = Line(Y, r_1)
a_2 = Line(C_1, m)
G_1 = Intersect(t_1, a_2)
b_2 = Segment(Y, G_1)
c_2 = Segment(D, G_1)
d_2 = Line(G, c_2)
H_1 = Intersect(d_2, b_2)
e_2 = Line(A_1, b_2)
f_2 = Line(H_1, g)
I_1 = Intersect(f_2, e_2)
g_2 = Line(I_1, s_1)
h_2 = Line(E_1, e_2)
J_1 = Intersect(g_2, h_2)
i_2 = Segment(I_1, J_1)
j_2 = Segment(J_1, E_1)
k_2 = Segment(E_1, A_1)
l_2 = Segment(A_1, I_1)
m_2 = Line(F_1, e_2)
n_2 = Line(H_1, g_2)
K_1 = Intersect(m_2, n_2)
p_2 = Segment(J_1, K_1)
q_2 = Segment(K_1, H_1)
r_2 = Segment(E_1, F_1)
s_2 = Segment(F_1, Y)
t_2 = Segment(H_1, I_1)
a_3 = Segment(K_1, F_1)
b_3 = Segment(H_1, Y)
c_3 = Segment(Y, A_1)
o = Distance(D, G)
T1: "hauteur = " + u
v: u (22 - 2 u) (18 - 2 u)
T2: "Volume = " + v + " cm³"
d_3 = Segment(D, L_1)
e_3 = Line(G, d_3)
M_1 = Intersect(xAxis, e_3)
N_1 = Mirror(L_1, M_1)
O_1 = Mirror(L_1, N_1)
f_3 = OrthogonalLine(O_1, xAxis)
P_1: (x(O_1), f(x(O_1)))
g_3 = Segment(O_1, P_1)
h_3 = OrthogonalLine(P_1, f_3)
Q_1 = Intersect(yAxis, h_3)
i_3 = Segment(Q_1, P_1)
j_3 = Segment(P_1, O_1)
R_1 = Point(f_3)
k_3 = OrthogonalLine(R_1, yAxis)
S_1 = Intersect(k_3, yAxis)
T_1 = Midpoint(S_1, R_1)
w = Vector(T_1, R_1)
z = Vector(T_1, S_1)
T3: "P (" + u + " ; " + v + " )"
W_1 = (6, 0)
Y_1 = Point(xAxis)
Z_1 = Point(xAxis)
A_2 = (2, 0)
F_2 = Point(xAxis)
G_2 = Point(xAxis)
H_2 = Point(xAxis)
I_2 = Point(xAxis)
J_2 = Point(xAxis)
B_2 = Point(xAxis)
D_2 = Point(yAxis)
E_2 = Midpoint(D_2, L_1)
K_2 = Midpoint(E_2, L_1)
l_3 = OrthogonalLine(K_2, yAxis)
m_3 = OrthogonalLine(B_2, l_3)
n_3 = OrthogonalLine(A_2, l_3)
p_3 = OrthogonalLine(F_2, l_3)
q_3 = OrthogonalLine(Z_1, l_3)
r_3 = OrthogonalLine(G_2, l_3)
s_3 = OrthogonalLine(W_1, l_3)
t_3 = OrthogonalLine(H_2, l_3)
a_4 = OrthogonalLine(Y_1, l_3)
b_4 = OrthogonalLine(I_2, l_3)
c_4 = OrthogonalLine(J_2, l_3)
L_2 = Intersect(l_3, m_3)
M_2 = Intersect(l_3, n_3)
N_2 = Intersect(l_3, p_3)
O_2 = Intersect(l_3, q_3)
P_2 = Intersect(l_3, r_3)
Q_2 = Intersect(l_3, s_3)
R_2 = Intersect(l_3, t_3)
S_2 = Intersect(l_3, a_4)
T_2 = Intersect(l_3, b_4)
U_2 = Intersect(c_4, l_3)
d_4 = Segment(B_2, L_2)
e_4 = Segment(A_2, M_2)
f_4 = Segment(F_2, N_2)
g_4 = Segment(Z_1, O_2)
h_4 = Segment(G_2, P_2)
i_4 = Segment(W_1, Q_2)
j_4 = Segment(H_2, R_2)
k_4 = Segment(Y_1, S_2)
l_4 = Segment(I_2, T_2)
m_4 = Segment(J_2, U_2)
T4: "x = " + u
V_2 = Point(v_1)
n_4 = OrthogonalLine(V_2, v_1)
W_2 = Point(n_4)
Y_2 = Midpoint(W_2, V_2)
p_4 = OrthogonalLine(Y_2, n_4)
Z_2 = Midpoint(V_2, L_1)
A_3 = Midpoint(Z_2, L_1)
B_3 = Midpoint(V_2, Z_2)
C_3 = Mirror(B_3, V_2)
D_3 = Mirror(V_2, C_3)
q_4 = OrthogonalLine(A_3, p_4)
r_4 = OrthogonalLine(Z_2, p_4)
s_4 = OrthogonalLine(B_3, p_4)
t_4 = OrthogonalLine(C_3, p_4)
a_5 = OrthogonalLine(D_3, p_4)
E_3 = Intersect(p_4, s_4)
F_3 = Intersect(p_4, r_4)
G_3 = Intersect(p_4, q_4)
H_3 = Intersect(t_4, p_4)
I_3 = Intersect(a_5, p_4)
b_5 = Segment(I_3, D_3)
c_5 = Segment(H_3, C_3)
d_5 = Segment(Y_2, V_2)
e_5 = Segment(E_3, B_3)
f_5 = Segment(F_3, Z_2)
g_5 = Segment(G_3, A_3)
C_2 = Point(l)
h_5 = OrthogonalLine(C_2, l)
J_3 = Intersect(q, h_5)
K_3 = Intersect(n, h_5)
L_3 = Intersect(i, h_5)
o_1 = Distance(K_3, L_3)
M_3 = Midpoint(C_2, J_3)
N_3 = Midpoint(J_3, K_3)
O_3 = Midpoint(K_3, L_3)
w_1 = Vector(O_3, L_3)
z_1 = Vector(O_3, K_3)
u_2 = Vector(N_3, K_3)
v_2 = Vector(N_3, J_3)
w_2 = Vector(M_3, C_2)
z_2 = Vector(M_3, J_3)
P_3 = Point(g)
i_5 = OrthogonalLine(P_3, g)
Q_3 = Intersect(i_5, p)
R_3 = Intersect(m, i_5)
S_3 = Intersect(j, i_5)
T_3 = Midpoint(Q_3, R_3)
U_3 = Midpoint(R_3, S_3)
V_3 = Midpoint(P_3, Q_3)
u_3 = Vector(T_3, R_3)
v_3 = Vector(T_3, Q_3)
w_3 = Vector(U_3, S_3)
z_3 = Vector(U_3, R_3)
u_4 = Vector(V_3, Q_3)
v_4 = Vector(V_3, P_3)
T18: "largeur = " + larg
T17: "longueur = " + long
W_3 = (-9.75, 7.75)
Y_3 = (-5, 7.75)
Z_3 = (-5, 7)
A_4 = (-9.75, 7)
P_4, w_4, y_3, z_4, a_6 = Polygon(W_3, Y_3, Z_3, A_4)
Q_4, a_7, b_6, c_6, d_6 = Polygon(A, B, C, D)
B_4 = Point(l_2)
j_5 = OrthogonalLine(B_4, v_1)
C_4 = Point(j_5)
D_4 = Point(t_2)
k_5 = OrthogonalLine(D_4, v_1)
E_4 = Point(k_5)
T19: "V(" + u + ") = " + v
T20: "V '(" + u + ") = " + f2
l_5 = Segment(Q, R)
F_4 = (9.6, -4.233333333333333)
G_4 = (9.066666666666666, -3.3333333333333335)
u_5 = Vector(F_4, G_4)
H_4 = (6.733333333333333, -1.7333333333333334)
I_4 = (10.066666666666666, -1.7333333333333334)
J_4 = (10.1, -3.3)
K_4 = (6.766666666666667, -3.3)
R_4, h_6, i_6, j_6, k_6 = Polygon(H_4, I_4, J_4, K_4)
f_6 = Function(f, 0, 9)