Volume Graph

45

24

A rectangular sheet of tin 45cm×24cm
 is to be made into a box without top,
by cutting off square from each corner
 and folding up the flaps.What should
 be the side of the square to be cut off
 so that the volume of the box is maximum? 

Problem

Solution

a project by  >> Benny Varghese,PGT Maths JNV Pathanamthitta

A = (-11.88, 4.9)
a_4 = Line(A, xAxis)
B_3 = Point(Circle[A, 7.5])
b = Segment(A, B_3)
c = OrthogonalLine(B_3, b)
C = Point(Circle[B_3, 4])
d = Segment(B_3, C)
e = OrthogonalLine(A, b)
f = OrthogonalLine(C, c)
D = Intersect(f, e)
g = 0.16666666666666666 (slider from 0 to 2, step 0.08333333333333333)
h = Circle(D, g)
k = Circle(C, g)
p = Circle(B_3, g)
q = Circle(A, g)
E, F = Intersect(h, f)
G, H = Intersect(q, e)
I = Intersect(a_4, b)
J, K = Intersect(q, a_4)
L_2, M = Intersect(k, f)
N, O = Intersect(k, c)
P = Intersect(c, d)
Q = Intersect(c, d)
R = Intersect(c, d)
S, T = Intersect(p, c)
i = Segment(E, K)
U, V_2 = Intersect(h, e)
j = Segment(U, N)
W = Intersect(a_4, b)
Z = Intersect(a_4, b)
A_1 = Intersect(a_4, b)
B_1, C_1 = Intersect(p, a_4)
l = Segment(M, B_1)
m = Segment(H, T)
n = Line(E, K)
D_1 = Intersect(xAxis, n)
r = Circle(D_1, 7.5 - (2 * g))
E_1, F_1 = Intersect(r, xAxis)
G_1 = (-2.16, 3.26)
s = Ray(F_1, G_1)
t = Circle(F_1, g)
c_1 = Circle(F_1, 4 - (2 * g))
a_1 = Line(M, B_1)
H_1 = Intersect(c_1, s)
b_1 = Line(D_1, s)
d_1 = Line(H_1, xAxis)
I_1 = Intersect(b_1, d_1)
J_1, K_1 = Intersect(t, a_1)
e_1 = Circle(D_1, g)
L_1, M_1 = Intersect(e_1, n)
f_1 = Circle(I_1, g)
g_1 = Circle(H_1, g)
h_1 = Line(H_1, a_1)
i_1 = Line(I_1, n)
N_1, O_1 = Intersect(f_1, i_1)
P_1, Q_1 = Intersect(g_1, h_1)
j_1 = Segment(N_1, L_1)
k_1 = Segment(N_1, P_1)
l_1 = Segment(P_1, J_1)
m_1 = Segment(J_1, L_1)
R_1 = Intersect(j, n)
S_1 = Intersect(j, a_1)
T_1 = Intersect(m, a_1)
U_1 = Intersect(m, n)
poly1, d_2, c_2, b_2, a_2 = Polygon(D, C, B_3, A)
poly2, u_1, r_1, s_1, t_1 = Polygon(U_1, R_1, S_1, T_1)
poly3, i_2, d_3, f_2, h_2 = Polygon(I_1, D_1, F_1, H_1)
poly4, n_1, l_2, d_4, i_3 = Polygon(N_1, L_1, D_1, I_1)
poly5, n_2, i_4, h_3, p_1 = Polygon(N_1, I_1, H_1, P_1)
poly6, l_3, j_2, f_3, d_5 = Polygon(L_1, J_1, F_1, D_1)
poly7, j_3, p_2, h_4, f_4 = Polygon(J_1, P_1, H_1, F_1)
poly8, e_2, r_2, s_2, m_2 = Polygon(E, R_1, S_1, M)
poly9, u, r_3, u_2, h_5 = Polygon(U, R_1, U_1, H)
poly10, s_3, n_3, t_2, t_3 = Polygon(S_1, N, T, T_1)
poly11, k_2, u_3, t_4, b_3 = Polygon(K, U_1, T_1, B_1)
poly12, d_6, e_3, r_4, u_4 = Polygon(D, E, R_1, U)
poly13, m_3, c_3, n_4, s_4 = Polygon(M, C, N, S_1)
poly14, t_5, t_6, b_4, b_5 = Polygon(T_1, T, B_3, B_1)
poly15, h_6, u_5, k_3, a = Polygon(H, U_1, K, A)
v: (6 * b)
w: (6 * d)
o: (6 * g)
poly16, d_7, a_3, b_6, c_4 = Polygon(D, A, B_3, C)
V: (o / 3, ((o * (45 - (2 * o))) * (24 - (2 * o)) / 316))
vol: (y(V) * 316)
text1: "Volume=" + (LaTeX[vol]) + ""
o_1 = false
V_1 = (-12.18, 4.6)
q_1 = Line(V_1, xAxis)
g_2 = OrthogonalLine(V_1, q_1)
W_1 = Point(q_1)
Z_1 = Point(q_1)
v_1 = Vector(W_1, Z_1)
A_2 = Point(q_1)
B_2 = Point(q_1)
w_1 = Vector(A_2, B_2)
C_2 = Point(g_2)
D_2 = Point(g_2)
z_1 = Vector(C_2, D_2)
E_2 = Point(g_2)
F_2 = Point(g_2)
v_2 = Vector(E_2, F_2)
G_2 = Intersect(c_2, t_6)
H_2 = Point(t_6)
text4: "⇐  side of square = " + o + ""
I_2 = Point(d_3)
J_2 = Point(f_2)
K_2 = Point(h_3)
L: v - (2 * o)
B: w - (2 * o)
text5: "L=" + (LaTeX[L]) + ""
text6: "B=" + (LaTeX[B]) + ""
text7: "H=" + (LaTeX[o]) + ""
z = false
o_2 = false