{{{id=86|
# Copyright 2009, P. Lutus
# Released under the GPL (http://www.gnu.org/copyleft/gpl.html)
///
}}}
{{{id=45|
%auto
# initialization
# reset() # commented out for now -- ticket 7255
var('f m b v x y z a b c d x_a x_b x_t y_a y_b y_t z_b z_t')
# special equation rendering
def render(x,name = "temp.png",size = "normal"):
if(type(x) != type("")): x = latex(x)
latex.eval("\\" + size + " $" + x + "$",{},"",name)
def radians(x):
return x * pi / 180
///
}}}
{{{id=79|
render(latex(v == (x_b+x_t)/2 * y * z),"full_volume_first_order_trapezoid.png","Large")
///
}}}
{{{id=73|
render("$y = mx+b$","regress.png","Large")
///
}}}
{{{id=74|
render(latex(m == (y_b-y_a)/(x_b-x_a)),"m_equ.png","Large")
///
}}}
{{{id=76|
render(latex(b == (y_b-y_a)*m),"b_equ.png","Large")
///
}}}
{{{id=75|
render("$f(x_a) = y_a$","temp.png","Large")
///
}}}
{{{id=80|
render("$f(x_b) = y_b$","temp.png","Large")
///
}}}
{{{id=49|
%auto
# interpolation function
ntrp(x,x_b,x_a,y_b,y_a) = (x-x_a)/(x_b-x_a)*(y_b-y_a)+y_a
render(y == ntrp(x,x_b,x_a,y_b,y_a),"ntrp.png","Large")
///
}}}
{{{id=81|
print " °c °f "
for c in range(0,110,10):
print "%4d %4.0f" % (c,ntrp(c,100,0,212,32))
///
°c °f
0 32
10 50
20 68
30 86
40 104
50 122
60 140
70 158
80 176
90 194
100 212
}}}
{{{id=63|
%auto
# first-order (x) trapezoidal tank area
fax(y,x_t,x_b,z,y_t,y_b) = ntrp(y,y_t,y_b,x_t,x_b) * z
render(a == fax(y,x_b,x_t,z,y_b,y_t),"fax.png","Large")
///
}}}
{{{id=88|
a == fax(y,x_t,x_b,z,y_t,y_b)
///
a == ((y - y_b)*(x_b - x_t)/(y_b - y_t) + x_b)*z
}}}
{{{id=83|
%auto
# first-order (x) trapezoidal tank volume
fvx(y,x_t,x_b,z,y_t,y_b) = integrate(fax(y,x_t,x_b,z,y_t,y_b),y)
render(v == fvx(y,x_t,x_b,z,y_t,y_b),"fvx.png","Large")
///
}}}
{{{id=46|
%auto
# second-order (x,z) trapezoidal tank area
faxz(y,x_t,x_b,z_t,z_b,y_t,y_b) = ntrp(y,y_t,y_b,x_t,x_b) * ntrp(y,y_t,y_b,z_t,z_b)
render(a == faxz(y,x_b,x_t,z_b,z_t,y_b,y_t),"faxz.png","Large")
///
}}}
{{{id=38|
%auto
# second-order (x,z) trapezoidal tank volume
fvxz(y,x_t,x_b,z_t,z_b,y_t,y_b) = integrate(faxz(y,x_t,x_b,z_t,z_b,y_t,y_b),y)
render(v == fvxz(y,x_t,x_b,z_t,z_b,y_t,y_b),"fvxz.png","Large")
///
}}}
{{{id=84|
fvxz(10,0,10,0,10,10,0)
///
1000/3
}}}
{{{id=87|
v == fvxz(y,x_t,x_b,z_t,z_b,y_t,y_b)
///
v == 1/6*(2*((x_b - x_t)*z_b - (x_b - x_t)*z_t)*y^3 - 3*(((2*x_b - x_t)*y_t - x_t*y_b)*z_b - ((x_b - 2*x_t)*y_b + x_b*y_t)*z_t)*y^2 + 6*((x_b*y_t^2 - x_t*y_b*y_t)*z_b - (x_b*y_b*y_t - x_t*y_b^2)*z_t)*y)/(y_b^2 - 2*y_b*y_t + y_t^2)
}}}
{{{id=85|
render("$v = fvxz(y,x_t,x_b,z_t,z_b,y_t,y_b)$","fvxz.png","Large")
///
}}}
{{{id=40|
html.table([[y,N(fvxz(y,10, 5, 12, 7,10,0))] for y in range(0,11)])
///
}}}
{{{id=77|
def charta(xx_b,xx_a,yy_b,yy_a):
p1 = plot(ntrp(x,xx_b,xx_a,yy_b,yy_a),x,0,10,ymin=0,ymax=10,linestyle='--')
pt1 = point((3,ntrp(3,xx_b,xx_a,yy_b,yy_a)),pointsize=24,rgbcolor='#008000')
pt2 = point((7,ntrp(7,xx_b,xx_a,yy_b,yy_a)),pointsize=24,rgbcolor='#008000')
lbl1 = text("$x_a,y_a$",(3,ntrp(3,xx_b,xx_a,yy_b,yy_a)+0.8),rgbcolor='black',fontsize=13)
lbl2 = text("$x_b,y_b$",(7,ntrp(7,xx_b,xx_a,yy_b,yy_a)+0.8),rgbcolor='black',fontsize=13)
lbl3 = text("$" + latex(y == m*x+b) + "$",(5,5),rgbcolor='black',fontsize=15,horizontal_alignment='left')
lbl4 = text("$" + latex(m == (y_b-y_a)/(x_b-x_a)) + "$",(5,3.5),rgbcolor='black',fontsize=15,horizontal_alignment='left')
lbl5 = text("$" + latex(b == (y_b-y_a)*m) + "$",(5,2),rgbcolor='black',fontsize=15,horizontal_alignment='left')
return p1+pt1+pt2+lbl1+lbl2+lbl3+lbl4+lbl5
show(charta(7,3,8,5),figsize=(4,3))
///
}}}
{{{id=60|
def chartb(xx_b,xx_a,yy_b,yy_a):
p1 = plot(ntrp(x,xx_b,xx_a,yy_b,yy_a),x,0,10,ymin=0,ymax=10,linestyle='--')
pt1 = point((3,ntrp(3,xx_b,xx_a,yy_b,yy_a)),pointsize=24,rgbcolor='#008000')
pt2 = point((7,ntrp(7,xx_b,xx_a,yy_b,yy_a)),pointsize=24,rgbcolor='#008000')
lbl1 = text("$x_a,y_a$",(3,ntrp(3,xx_b,xx_a,yy_b,yy_a)+0.8),rgbcolor='black',fontsize=13)
lbl2 = text("$x_b,y_b$",(7,ntrp(7,xx_b,xx_a,yy_b,yy_a)+0.8),rgbcolor='black',fontsize=13)
lbl3 = text("$y = " + latex(ntrp(x,x_t,x_b,y_t,y_b)) + "$",(5,3),rgbcolor='black',fontsize=15)
return p1+pt1+pt2+lbl1+lbl2+lbl3
show(chartb(7,3,8,5),figsize=(4,3))
///
}}}
{{{id=62|
def tank3d(x_t,x_b,y_t,y_b,z_t,z_b):
th = 2
lines = 0
# verticals
lines += line3d([[0,y_b,0],[0,y_t,0]],thickness=th)
lines += line3d([[0,y_b,z_b],[0,y_t,z_t]],thickness=th)
lines += line3d([[x_b,y_b,0],[x_t,y_t,0]],thickness=th)
lines += line3d([[x_b,y_b,z_b],[x_t,y_t,z_t]],thickness=th)
# bottom
lines += line3d([[0,y_b,0],[x_b,y_b,0],[x_b,y_b,z_b],[0,y_b,z_b],[0,y_b,0]],thickness=th)
# top
lines += line3d([[0,y_t,0],[x_t,y_t,0],[x_t,y_t,z_t],[0,y_t,z_t],[0,y_t,0]],thickness=th)
lines += text3d("xb,yb,zb",(x_b,y_b-2,z_b))
lines += text3d("xt,yt,zt",(x_t+5,y_t,z_t))
lines = lines.rotateX(radians(-73))
lines = lines.rotateZ(radians(-60))
lines = lines.rotateX(radians(3))
return lines
///
}}}
{{{id=66|
bb = tank3d(40, 25, 40,0,90, 60)
show(bb,frame=False,aspect_ratio=1,figsize=[4,3])
///
}}}
{{{id=72|
///
}}}