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Bodies with fixed volume and minimal surface

Author: Valery Ochkov (http://twt.mpei.ac.ru/ochkov/v_ochkov.htm)

> 

restart

1. Cylinder with a bottom
 

Volume (V)

> 

h__1 := solve(V = (1/4)*Pi*d^2*h, h)

4*V/(Pi*d^2)

 (1.1)

Surface (S as function of V and d)

> 

S := subs(h = h__1, (1/4)*Pi*d^2+Pi*d*h)

(1/4)*Pi*d^2+4*V/d

 (1.2)
> 

d__optimal := solve(diff(S, d), d)

2*(V*Pi^2)^(1/3)/Pi, -(V*Pi^2)^(1/3)/Pi+I*3^(1/2)*(V*Pi^2)^(1/3)/Pi, -(V*Pi^2)^(1/3)/Pi-I*3^(1/2)*(V*Pi^2)^(1/3)/Pi

 (1.3)

A = d / h if S is minimal

> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

2

 (1.4)

2. Cylinder with a bottom (see above) and a cover
 
> 

h__1 := solve(V = (1/4)*Pi*d^2*h, h)

4*V/(Pi*d^2)

 (2.1)

Surface (S as function of V and d)

> 

S := subs(h = h__1, (1/4)*Pi*d^2+Pi*d*h+(1/4)*Pi*d^2)

(1/2)*Pi*d^2+4*V/d

 (2.2)
> 

d__optimal := solve(diff(S, d), d)

4^(1/3)*(V*Pi^2)^(1/3)/Pi, -(1/2)*4^(1/3)*(V*Pi^2)^(1/3)/Pi+((1/2)*I)*3^(1/2)*4^(1/3)*(V*Pi^2)^(1/3)/Pi, -(1/2)*4^(1/3)*(V*Pi^2)^(1/3)/Pi-((1/2)*I)*3^(1/2)*4^(1/3)*(V*Pi^2)^(1/3)/Pi

 (2.3)

A = d / h if S is minimal

> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

1

 (2.4)

3. Cylinder with a bottom and one central vertical dividing wall
 
> 

h__1 := solve(V = (1/4)*Pi*d^2*h, h)

4*V/(Pi*d^2)

 (3.1)
> 

S := subs(h = h__1, (1/4)*Pi*d^2+Pi*d*h+d*h)

(1/4)*Pi*d^2+4*V/d+4*V/(d*Pi)

 (3.2)
> 

d__optimal := solve(diff(S, d), d)

2*(V*(Pi+1)*Pi)^(1/3)/Pi, -(V*(Pi+1)*Pi)^(1/3)/Pi+I*3^(1/2)*(V*(Pi+1)*Pi)^(1/3)/Pi, -(V*(Pi+1)*Pi)^(1/3)/Pi-I*3^(1/2)*(V*(Pi+1)*Pi)^(1/3)/Pi

 (3.3)
> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

(2*Pi+2)/Pi

 (3.4)
> 

evalf(A)

2.636619772

 (3.5)

4. Cone
 

> 

h__1 := solve(V = ((1/3)*Pi*((1/4)*d^2))*h, h)

12*V/(Pi*d^2)

 (4.1)
> 

S := subs(h = h__1, (1/2)*Pi*d*sqrt((1/4)*d^2+h^2))

(1/4)*Pi*d*(d^2+576*V^2/(Pi^2*d^4))^(1/2)

 (4.2)
> 

d__optimal := solve(diff(S, d), d)

288^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (1/2+((1/2)*I)*3^(1/2))*288^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (-1/2+((1/2)*I)*3^(1/2))*288^(1/6)*(V^2*Pi^4)^(1/6)/Pi, -288^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (-1/2-((1/2)*I)*3^(1/2))*288^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (1/2-((1/2)*I)*3^(1/2))*288^(1/6)*(V^2*Pi^4)^(1/6)/Pi

 (4.3)
> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

2^(1/2)*csgn(V)

 (4.4)
> 

evalf(A)

1.414213562*csgn(V)

 (4.5)

5. Cone with base (a wineglass)
 
> 

``

> 

h__1 := solve(V = ((1/3)*Pi*((1/4)*d^2))*h, h)

12*V/(Pi*d^2)

 (5.1)
> 

S := subs(h = h__1, (1/2)*Pi*d*sqrt(((1/2)*d)^2+h^2)+(1/4)*Pi*d^2)

(1/4)*Pi*d*(d^2+576*V^2/(Pi^2*d^4))^(1/2)+(1/4)*Pi*d^2

 (5.2)
> 

d__optimal := solve(diff(S, d), d)

72^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (1/2+((1/2)*I)*3^(1/2))*72^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (-1/2+((1/2)*I)*3^(1/2))*72^(1/6)*(V^2*Pi^4)^(1/6)/Pi, -72^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (-1/2-((1/2)*I)*3^(1/2))*72^(1/6)*(V^2*Pi^4)^(1/6)/Pi, (1/2-((1/2)*I)*3^(1/2))*72^(1/6)*(V^2*Pi^4)^(1/6)/Pi

 (5.3)
> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

(1/2)*2^(1/2)*csgn(V)

 (5.4)
> 

evalf(A)

.7071067810*csgn(V)

 (5.5)

6. Cone with base (see above) and cover (a cap)
 
> 

h__1 := solve(V = ((1/3)*Pi*((1/4)*d^2))*h, h)

12*V/(Pi*d^2)

 (6.1)
> 

S := subs(h = h__1, (1/2)*Pi*d*sqrt(((1/2)*d)^2+h^2)+(1/4)*Pi*d^2+(1/4)*Pi*d^2)

(1/4)*Pi*d*(d^2+576*V^2/(Pi^2*d^4))^(1/2)+(1/2)*Pi*d^2

 (6.2)
> 

d__optimal := solve(diff(S, d), d)

96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, (1/2+((1/2)*I)*3^(1/2))*96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, (-1/2+((1/2)*I)*3^(1/2))*96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, -96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, (-1/2-((1/2)*I)*3^(1/2))*96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, (1/2-((1/2)*I)*3^(1/2))*96^(1/6)*(Pi^4*V^2*(2*7^(1/2)-5))^(1/6)/Pi, 96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi, (1/2+((1/2)*I)*3^(1/2))*96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi, (-1/2+((1/2)*I)*3^(1/2))*96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi, -96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi, (-1/2-((1/2)*I)*3^(1/2))*96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi, (1/2-((1/2)*I)*3^(1/2))*96^(1/6)*(-Pi^4*V^2*(2*7^(1/2)+5))^(1/6)/Pi

 (6.3)
> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

(1/3)*6^(1/2)*(2*7^(1/2)-5)^(1/2)*csgn(V)

 (6.4)
> 

evalf(A)

.4408345283*csgn(V)

 (6.5)

7. Cone with semisphere - Can you get the solution? And for others bodies!
 
> 

NULL

> 

h__1 := solve(V = ((1/3)*Pi*((1/4)*d^2))*h+(4/3)*Pi*((1/2)*d)^3*(1/2), h)

-(Pi*d^3-12*V)/(Pi*d^2)

 (7.1)
> 

S := subs(h = h__1, (1/2)*Pi*d*sqrt(((1/2)*d)^2+h^2)+(1/2)*Pi*d^2)

(1/4)*Pi*d*(d^2+4*(Pi*d^3-12*V)^2/(Pi^2*d^4))^(1/2)+(1/2)*Pi*d^2

 (7.2)
> 

d__optimal := solve(diff(S, d), d)

RootOf(5*Pi^4*_Z^12+144*Pi^3*V*_Z^9-4608*Pi^2*V^2*_Z^6+13824*Pi*V^3*_Z^3+82944*V^4)

 (7.3)
> 

d__optimal := allvalues(d__optimal)

(1/5)*100^(1/3)*(V*(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)-9)*Pi^2)^(1/3)/Pi, -(1/10)*100^(1/3)*(V*(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)-9)*Pi^2)^(1/3)/Pi+((1/10)*I)*3^(1/2)*100^(1/3)*(V*(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)-9)*Pi^2)^(1/3)/Pi, -(1/10)*100^(1/3)*(V*(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)-9)*Pi^2)^(1/3)/Pi-((1/10)*I)*3^(1/2)*100^(1/3)*(V*(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)-9)*Pi^2)^(1/3)/Pi, (1/5)*(-100*V*(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)+9)*Pi^2)^(1/3)/Pi, -(1/10)*(-100*V*(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)+9)*Pi^2)^(1/3)/Pi-((1/10)*I)*3^(1/2)*(-100*V*(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)+9)*Pi^2)^(1/3)/Pi, -(1/10)*(-100*V*(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)+9)*Pi^2)^(1/3)/Pi+((1/10)*I)*3^(1/2)*(-100*V*(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)-214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)+9)*Pi^2)^(1/3)/Pi, (1/5)*100^(1/3)*(V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-9)*Pi^2)^(1/3)/Pi, -(1/10)*100^(1/3)*(V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-9)*Pi^2)^(1/3)/Pi+((1/10)*I)*3^(1/2)*100^(1/3)*(V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-9)*Pi^2)^(1/3)/Pi, -(1/10)*100^(1/3)*(V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-9)*Pi^2)^(1/3)/Pi-((1/10)*I)*3^(1/2)*100^(1/3)*(V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-9)*Pi^2)^(1/3)/Pi, (1/5)*(-100*V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)+3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+9)*Pi^2)^(1/3)/Pi, -(1/10)*(-100*V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)+3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+9)*Pi^2)^(1/3)/Pi-((1/10)*I)*3^(1/2)*(-100*V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)+3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+9)*Pi^2)^(1/3)/Pi, -(1/10)*(-100*V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)+3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+9)*Pi^2)^(1/3)/Pi+((1/10)*I)*3^(1/2)*(-100*V*(((-5*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)*((18*I)*503^(1/2)+253)^(2/3)+1032*3^(1/2)*((18*I)*503^(1/2)+253)^(1/3)+214*((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-305*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))/(((18*I)*503^(1/2)+253)^(1/3)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)))^(1/2)*3^(1/2)+3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+9)*Pi^2)^(1/3)/Pi

 (7.4)
> 

A := simplify(d__optimal/(eval(h__1, d = d__optimal)))

(-3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)-(-3*(5*((18*I)*503^(1/2)+253)^(2/3)-214*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3)-3096*3^(1/2)/((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))^(1/2)+9)/(3^(1/2)*((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2)+(-3*(5*((18*I)*503^(1/2)+253)^(2/3)-214*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3)-3096*3^(1/2)/((5*((18*I)*503^(1/2)+253)^(2/3)+107*((18*I)*503^(1/2)+253)^(1/3)+305)/((18*I)*503^(1/2)+253)^(1/3))^(1/2))^(1/2)-24)

 (7.5)
> 

evalf(A)

-4.778021643-0.3797578066e-8*I

 (7.6)
> 

``