{{{id=159|
%auto
reset()
forget()
///
}}}
{{{id=369|
f(y) = 2 * sqrt((2-y)*y)
ay = []
ai = []
n = 400
for i in range(n+1):
y = i*2/n
ai += [y]
ay += [f(y)]
p = list_plot(zip(ay,ai),figsize=(2.5,4),axes_labels=['Width (w)','Height (y)'],plotjoined=True)
p += text('$w = 2\ \sqrt{(2-y)y}$',(1,1),fontsize=12,color='black')
save(p,'/netbackup/data/Network/arachnoid/TankCalc/volumes_resources/cylindrical_tank_equation_plot.png')
show(p)
///
}}}
{{{id=371|
forget()
var('y,z')
assume(y > 0)
fi(y) = integrate(f(z),(z,0,y))
show(f(y))
show(fi(y))
///
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \sqrt{-{\left(y - 2\right)} y}
\newcommand{\Bold}[1]{\mathbf{#1}}\pi + \sqrt{-y + 2} {\left(y - 1\right)} \sqrt{y} - 2 \, \arctan\left(\frac{\sqrt{-y + 2}}{\sqrt{y}}\right)
}}}
{{{id=372|
fi(2)
///
pi
}}}
{{{id=373|
p2 = plot(fi(y),(y,0,2),figsize=(3,3),axes_labels=['Height (y)','Area (a)'])
p2 += text('$a = ' + latex(fi(y)) + '$',(1.5,1.5),fontsize=11,color='black')
save(p2,'/netbackup/data/Network/arachnoid/TankCalc/volumes_resources/cylindrical_tank_integral_plot.png')
show(p2)
///
}}}
{{{id=378|
# bucket (conical frustum) equation
var('r_a,r_b,h_b,h_c')
show(latex(1/3 * pi * (r_b^2 * h_c - r_a^2 * (h_c-h_b))))
h_c = h_b * r_b / (r_b-r_a)
vv = 1/3 * pi * r_b^2 * h_c - 1/3 * pi * r_a^2 * (h_c-h_b)
bv(h_b,r_b,r_a) = vv
show(bv(h_b,r_b,r_a))
show(latex(bv(h_b,r_b,r_a)))
latex(vv)
vvfs = vv.full_simplify()
show(vvfs)
latex(vvfs)
///
\newcommand{\Bold}[1]{\mathbf{#1}}\hbox{\frac{1}{3} \, {\left({\left(h_{b} - h_{c}\right)} r_{a}^{2} + h_{c} r_{b}^{2}\right)} \pi}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\pi h_{b} r_{b}^{3}}{3 \, {\left(r_{a} - r_{b}\right)}} + \frac{1}{3} \, {\left(\frac{h_{b} r_{b}}{r_{a} - r_{b}} + h_{b}\right)} \pi r_{a}^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\hbox{-\frac{\pi h_{b} r_{b}^{3}}{3 \, {\left(r_{a} - r_{b}\right)}} + \frac{1}{3} \, {\left(\frac{h_{b} r_{b}}{r_{a} - r_{b}} + h_{b}\right)} \pi r_{a}^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{3} \, \pi h_{b} r_{a}^{2} + \frac{1}{3} \, \pi h_{b} r_{a} r_{b} + \frac{1}{3} \, \pi h_{b} r_{b}^{2}
\frac{1}{3} \, \pi h_{b} r_{a}^{2} + \frac{1}{3} \, \pi h_{b} r_{a} r_{b} + \frac{1}{3} \, \pi h_{b} r_{b}^{2}
}}}
{{{id=379|
show(bv(h_b,r_b,r_a))
latex(bv(h_b,r_b,r_a))
///
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\pi h_{b} r_{b}^{3}}{3 \, {\left(r_{a} - r_{b}\right)}} + \frac{1}{3} \, {\left(\frac{h_{b} r_{b}}{r_{a} - r_{b}} + h_{b}\right)} \pi r_{a}^{2}
-\frac{\pi h_{b} r_{b}^{3}}{3 \, {\left(r_{a} - r_{b}\right)}} + \frac{1}{3} \, {\left(\frac{h_{b} r_{b}}{r_{a} - r_{b}} + h_{b}\right)} \pi r_{a}^{2}
}}}
{{{id=380|
bv(10,10,9.9999999)
///
999.999988555908*pi
}}}
{{{id=389|
# bucket (conical frustum) incremental volume equation
var('r_a,r_b,y,h,q')
vf(r_a,r_b,h,y) = pi * (r_a + (r_b - r_a)*(y/h))^2
vi(r_a,r_b,h,y) = integrate(pi * (r_a + (r_b - r_a)*(q/h))^2, (q, 0, y))
show(latex(vf(r_a,r_b,h,y)))
show(vf(r_a,r_b,h,y))
show(latex(vi(r_a,r_b,h,y)))
show(vi(r_a,r_b,h,y))
print(vi(r_a,r_b,h,y))
///
\newcommand{\Bold}[1]{\mathbf{#1}}\hbox{{\left(\frac{{\left(r_{a} - r_{b}\right)} y}{h} - r_{a}\right)}^{2} \pi}
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{{\left(r_{a} - r_{b}\right)} y}{h} - r_{a}\right)}^{2} \pi
\newcommand{\Bold}[1]{\mathbf{#1}}\hbox{\frac{{\left(3 \, h^{2} r_{a}^{2} y + {\left(r_{a}^{2} - 2 \, r_{a} r_{b} + r_{b}^{2}\right)} y^{3} - 3 \, {\left(h r_{a}^{2} - h r_{a} r_{b}\right)} y^{2}\right)} \pi}{3 \, h^{2}}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(3 \, h^{2} r_{a}^{2} y + {\left(r_{a}^{2} - 2 \, r_{a} r_{b} + r_{b}^{2}\right)} y^{3} - 3 \, {\left(h r_{a}^{2} - h r_{a} r_{b}\right)} y^{2}\right)} \pi}{3 \, h^{2}}
1/3*(3*h^2*r_a^2*y + (r_a^2 - 2*r_a*r_b + r_b^2)*y^3 - 3*(h*r_a^2 - h*r_a*r_b)*y^2)*pi/h^2
}}}
{{{id=390|
vi(5,10,10,1)
///
331/12*pi
}}}
{{{id=391|
# test results
for i in range(11):
print '%8.2f %8.2f' % (i,vi(5,10,10,i))
///
0.00 0.00
1.00 86.66
2.00 190.59
3.00 313.37
4.00 456.58
5.00 621.77
6.00 810.53
7.00 1024.42
8.00 1265.01
9.00 1533.88
10.00 1832.60
}}}
{{{id=392|
# compare to original equation
print(N(bv(1,5.5,5)))
print(N(bv(5,7.5,5)))
print(N(bv(10,10,5)))
///
86.6555973615184
621.773546022980
1832.59571459405
}}}
{{{id=394|
# now the inverse case -- height for volume
var('r_a,r_b,h,v,y')
yf(r_a,r_b,h,v) = solve(vi(r_a,r_b,h,y) - v == 0,y)[2].rhs()
///
}}}
{{{id=396|
show(yf(r_a,r_b,h,v))
show(latex(yf(r_a,r_b,h,v)))
print(yf(r_a,r_b,h,v))
///
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{h r_{a}}{r_{a} - r_{b}} - \frac{{\left(\pi h r_{a}^{3} - 3 \, {\left(r_{a} - r_{b}\right)} v\right)}^{\left(\frac{1}{3}\right)} {\left(h^{2}\right)}^{\left(\frac{1}{3}\right)}}{{\left(r_{a} - r_{b}\right)} \pi^{\left(\frac{1}{3}\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}\hbox{\frac{h r_{a}}{r_{a} - r_{b}} - \frac{{\left(\pi h r_{a}^{3} - 3 \, {\left(r_{a} - r_{b}\right)} v\right)}^{\left(\frac{1}{3}\right)} {\left(h^{2}\right)}^{\left(\frac{1}{3}\right)}}{{\left(r_{a} - r_{b}\right)} \pi^{\left(\frac{1}{3}\right)}}}
h*r_a/(r_a - r_b) - (pi*h*r_a^3 - 3*(r_a - r_b)*v)^(1/3)*(h^2)^(1/3)/((r_a - r_b)*pi^(1/3))
}}}
{{{id=397|
# generate some test results -- create volume from height, then height from volume
import re
txt = ''
html = ''
for i in range(1,11):
vv = vi(5,10,10,i)
yy = yf(5,10,10,vv)
s = '%10.4f %10.4f %10.4f\n' % (i,vv,yy)
txt += s
hs = '' + re.sub('\s*(\S+)\s*','| \\1 | ',s) + '
\n'
html += hs
print (txt)
print (html)
///
1.0000 86.6556 1.0000
2.0000 190.5900 2.0000
3.0000 313.3739 3.0000
4.0000 456.5781 4.0000
5.0000 621.7735 5.0000
6.0000 810.5309 6.0000
7.0000 1024.4210 7.0000
8.0000 1265.0146 8.0000
9.0000 1533.8826 9.0000
10.0000 1832.5957 10.0000
| 1.0000 | 86.6556 | 1.0000 |
| 2.0000 | 190.5900 | 2.0000 |
| 3.0000 | 313.3739 | 3.0000 |
| 4.0000 | 456.5781 | 4.0000 |
| 5.0000 | 621.7735 | 5.0000 |
| 6.0000 | 810.5309 | 6.0000 |
| 7.0000 | 1024.4210 | 7.0000 |
| 8.0000 | 1265.0146 | 8.0000 |
| 9.0000 | 1533.8826 | 9.0000 |
| 10.0000 | 1832.5957 | 10.0000 |
}}}
{{{id=400|
///
}}}