Ronan

Would this answer be of use?...

Answered: Ronan 1341

November 06 2024

1 9

I answered something similar in 2016. I don't have the time right now to investicate you example.

How to take derivative of sum? - MaplePrimes

Mentally...

Answered: Ronan 1341

October 29 2024

1 0

10:55-10:10 =45
11:58-10:55=63
9 divides 45 and 63

so 9

align and \n...

Answered: Ronan 1341

September 22 2024

2 1

Use align and \n (within the text) as an option in textplot or textplot3d. This is just a quick example.

with(plots);with(plottools)
A:=[0,0,0];C[1]:=[1,1,1]
plt1:=point([A,C[1]],symbol=solidcircle,symbolsize=16,color=blue)
plt2:=textplot3d([ [A[],"A",align={above,right}] , [C[1][],typeset("C[1]=\n",C[1]),align={below,left}] ]):

display(plt1,plt2)

Like this...

Answered: Ronan 1341

September 17 2024

0 0

One way of doing it.

>

restart

>

local D;

>	BD:=4; AB:=6;

(1)

>	eq1:=BDsin(4theta)=ABsin(2theta)

(2)

>	sol:=solve(eq1,[theta])

[[theta = 0], [theta = (1/2)*Pi], [theta = (1/2)*arctan((1/3)*7^(1/2))], [theta = -(1/2)*arctan((1/3)*7^(1/2))]]

(3)

>	assign(sol[3]); theta:=simplify(theta)

(4)

>	eq2:=ABsin(2theta)=BC*sin(theta)

(5)

>	#eq3:=eval(eq2,(sol[3]))

>

>	simplify(solve(eq2,[BC])[])

(6)

>

assign(%)

>

BC

(7)

>	DC:=BDcos(Pi-4theta)+BC*cos(theta)

-1/2+12*cos((1/4)*arctan(3*7^(1/2)))^2

(8)

>	simplify( (8) );

(9)

Download 2024-09-18_A_Find_DC_geometry.mw

Like this....

Answered: Ronan 1341

September 17 2024

1 0

180-2m+60+k = 2n+40+k implies m+n=100 so m=100-n

m+40+n+x=180

100-n+40+n+x=180

therefore x=40

Doing it visually...

Answered: Ronan 1341

September 13 2024

3 2

Looking at it.ABC is half the area and ABE is quater the area. Draw a vertical line from E intersects the diagonal midpoint at say G. Triangle ABF and EGF are similar. with a 4;1 area ratio. So the horizontal coordinate of F is 2/3 AE. also ABF+EGF is quater of the total area.

The triangle EGC area. is 1/8 of the total area so 3cm^2.

ABF+2* EGF would be 6 (quater of the total area)

EGF would be 1/6 of that =1. So total area of 4.

Suggestion:-One Drive or Google docs...

Answered: Ronan 1341

August 20 2024

0 3

Could you put the zip file on One Drive Google Docs or Dropbox for example and have a link to it it in your post.

A proof...

Answered: Ronan 1341

June 24 2024

0 9

I got curious. I have converted away from the geometry package and used a rational parameterisation for cos and sin

[a cos(t), b sin(t)] to [a* (1t^2)/(1+t^2), b*2*t/(1+t^2)] . In this case t is not the angle.

This proves the angle is constant upto sign changes for signum . But that just changes the reported angle from sfor example 50deg to 130deg.

I would like to know the name of this theorem.

>	restart; with(plots):with(LinearAlgebra):

>

x0 := 100;
y0 := 40;
a := 7;
b := 5;
c := sqrt(a^2 - b^2);
e1:= x^2/a^2 + y^2/b^2 - 1;
F1:=[ -c, 0];
F2:=[ c, 0];
eq := simplify((a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*x0*y0*(x - x0)*(y - y0)) = 0;
sol := solve({eq}, {y});
tang1:=(lhs-rhs)~(op(sol[1]));

tang2:= (lhs-rhs)~(op(sol[2]));
sol2 := op(solve({op(sol[1]), x^2/a^2 + y^2/b^2 - 1 }, {x, y}));
xM2 := rhs(sol2[1]);
yM2 := rhs(sol2[2]);
A:=[ xM2, yM2];
sol3 := op(solve({op(sol[2]), x^2/a^2 + y^2/b^2 - 1 }, {x, y}));
xM3 := rhs(sol3[1]);
yM3 := rhs(sol3[2]);
B:=[ xM3, yM3];

slpvecAB:=<(B[2]-A[2]),B[1]-A[1]>; #slpvec
Pol:=-slpvecAB[1]*(x-A[1])+slpvecAB[2]*(y-A[2]);
simplify(Pol);
isolate(%, y);

TANG := proc(t) local xM, yM; xM :=a*(1-t^2)/(1+t^2);# a*cos(t);
               yM :=b*2*t/(1+t^2);# b*sin(t);
               return(expand(1/49*x*xM + 1/25*y*yM - 1 ));
        end proc;

#t := -0.25;

TT:= TANG(t);
#Equation(TT);
M:=[a*(1-t^2)/(1+t^2),b*2*t/(1+t^2)];#[ a*cos(t), b*sin(t)];

${y = (4000/9951)*x-1960/9951+(5/9951)*13087^(1/2)*x-(500/9951)*13087^(1/2)}, {y = (4000/9951)*x-1960/9951-(5/9951)*13087^(1/2)*x+(500/9951)*13087^(1/2)}$

$slpvecAB := Vector(2, {(1) = (125/1642)*13087^(1/2), (2) = -(49/821)*13087^(1/2)})$

proc (t) local xM, yM; xM := a*(1-t^2)/(1+t^2); yM := 2*b*t/(1+t^2); return expand((1/49)*x*xM+(1/25)*y*yM-1) end proc

-(1/7)*x*t^2/(t^2+1)+(1/7)*x/(t^2+1)+(2/5)*y*t/(t^2+1)-1

[7*(-t^2+1)/(t^2+1), 10*t/(t^2+1)]

(1)

>

>	P:=rhs~(solve( [tang1, TT],[x,y])[]);

[-7*(93*t+56+13087^(1/2))/(93*t-56-13087^(1/2)), -(5/107)*(56+13087^(1/2))*(107*t-13087^(1/2)+56)/(93*t-56-13087^(1/2))]

(2)

>	Q:=rhs~(solve( [tang2, TT],[x,y])[]);

[-7*(93*t+56-13087^(1/2))/(93*t-56+13087^(1/2)), (5/107)*(-56+13087^(1/2))*(107*t+13087^(1/2)+56)/(93*t-56+13087^(1/2))]

(3)

>	slpvecPF2:=simplify(<(P[2]-F2[2]),P[1]-F2[1]>); PF2:=simplify(-slpvecPF2[1](x-P[1])+slpvecPF2[2](y-P[2]));

>

$slpvecPF2 := Vector(2, {(1) = -5*(56+13087^(1/2))*(-107*t+13087^(1/2)-56)/(-9951*t+5992+107*13087^(1/2)), (2) = ((-186*t+2*13087^(1/2)+112)*6^(1/2)-651*t-7*13087^(1/2)-392)/(93*t-56-13087^(1/2))})$

(2*(465+(-5*t+y)*13087^(1/2)+(-93*y-280)*t+56*y)*6^(1/2)+(5*t*x-7*y)*13087^(1/2)+7*(40*x-93*y)*t-465*x-392*y)/(93*t-56-13087^(1/2))

(4)

>	slpvecQF2:=simplify(<(Q[2]-F2[2]),Q[1]-F2[1]>); QF2:=simplify(-slpvecQF2[1](x-Q[1])+slpvecQF2[2](y-Q[2]));

$slpvecQF2 := Vector(2, {(1) = 5*(-56+13087^(1/2))*(107*t+13087^(1/2)+56)/(9951*t-5992+107*13087^(1/2)), (2) = ((-2*6^(1/2)+7)*13087^(1/2)+(-186*t+112)*6^(1/2)-651*t-392)/(93*t-56+13087^(1/2))})$

(((10*t-2*y)*13087^(1/2)+(-186*y-560)*t+112*y+930)*6^(1/2)+(-5*t*x+7*y)*13087^(1/2)+(280*x-651*y)*t-465*x-392*y)/(93*t-56+13087^(1/2))

(5)

>

>	alpha :=(simplify(VectorAngle(slpvecQF2,slpvecPF2),assume=real))

(1/2)*Pi-signum(93*t-56+13087^(1/2))*signum(93*t-56-13087^(1/2))*arcsin((93/14)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)))

(6)

>	Setof:=frontend(indets,[alpha])

${Pi, arcsin((93/14)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2))), signum(93*t-56-13087^(1/2)), signum(93*t-56+13087^(1/2))}$

(7)

>	(sin(Setof[2]))^2

(8649/196)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)^2/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018))

(8)

>	simpascn:=arcsin(sqrt(factor( (8) ))); # the t's eliminate

(9)

>	alpha1:=(subs(Setof[2]=simpascn,alpha))*180/Pi

180*((1/2)*Pi-signum(93*t-56+13087^(1/2))*signum(93*t-56-13087^(1/2))*arcsin((1/58693852)*(1451954478752954-68582717957500*6^(1/2))^(1/2)))/Pi

(10)

>	plot(alpha1,t=-10..10,view=[-10..10,0..180])

>

Download 2024-06-24_constant_angle.mw

One possible way...

Answered: Ronan 1341

June 15 2024

2 0

Hope this helps. There are probably more efficient ways.

Download 2024-06-15_A_Binonm_coeffs.mw

How about using the Backup...

Answered: Ronan 1341

May 15 2024

0 5

Click in Restore Backup and list the files by date saved,

Use fsolve...

Answered: Ronan 1341

May 13 2024

1 0

The set of equations; as far as I can see, cannot be solved explicitly do to the trig. functions.
I rearranged your equations so you only need to solve eq5 and eq6 for y and z.

I randomly made up values for t, T, a,c,e,f just to show the use of fsolve.

Hope this helps you.

Edit I re-attached the worksheet because I had a mistake in it.

>

restart

>	t:=Pi/2; T:=t;

(1)

>	a:=3;c:=2;e:=4;f:=2

(2)

>

#z:=1

>

w := sqrt(a^2 - (a - y)^2);
v := cot(t + arcsin(w/a)) ;
u := (sqrt(c^2 - (c - z)^2) - v*x - w)/x^2;
x := y + z + e;
eq5 := v + cot(T + arcsin((u*x^2 + v*x + w)/c)) + 2*u*x = 0;
eq6 :=f = Pi/30*(6*u^2*x^5 + 15*u*v*x^4 + (20*u*w + 10*v^2)*x^3 + 30*v*w*x^2 + 30*x*w^2) - Pi/3*(-z^3 + 3*z^2*c - y^3 + 3*y^2*a);

-(9-(3-y)^2)^(1/2)/((3-y)^2)^(1/2)

((4-(2-z)^2)^(1/2)+(9-(3-y)^2)^(1/2)*x/((3-y)^2)^(1/2)-(9-(3-y)^2)^(1/2))/x^2

-(9-(3-y)^2)^(1/2)/((3-y)^2)^(1/2)-(4-(2-z)^2)^(1/2)/((2-z)^2)^(1/2)+2*((4-(2-z)^2)^(1/2)+(9-(3-y)^2)^(1/2)*(y+z+4)/((3-y)^2)^(1/2)-(9-(3-y)^2)^(1/2))/(y+z+4) = 0

(3)

>	fsolve({ eq5,eq6},complex);

(4)

>

Download 2024-05-13_A_An_approach.mw

One way...

Answered: Ronan 1341

April 01 2024

0 0

f:=x-> (1-k*x)/(1+x^2)
f := proc (x) options operator, arrow; (1-k*x)/(1+x^2) end proc

slpf:=diff(f(x),x)
                           k      2 (-k x + 1) x
                slp := - ------ - --------------
                          2                 2   
                         x  + 1     / 2    \    
                                    \x  + 1/    

m:=eval(slpf,x=3);# slope of tangent line at x=3

                              2      3 
                         m := -- k - --
                              25     50

f(3)
                            3      1 
                          - -- k + --
                            10     10

line:=unapply(m*x+c,x)
 line := proc (x) options operator, arrow; ((2/25)*k-3/50)*x+c 

    end proc




line(3)=f(3);# x=3 is tangent point
                  6      9          3      1 
                  -- k - -- + c = - -- k + --
                  25     50         10     10

c:=solve(line(3)=f(3),c)
                               27     7 
                        c := - -- k + --
                               50     25

line(x)
                   /2      3 \     27     7 
                   |-- k - --| x - -- k + --
                   \25     50/     50     25

Check under Options Display...

Answered: Ronan 1341

March 24 2024

3 1

Check under Tools Options Display Typesetting level to see if it is set to extended.

It looks like I have it working correctl...

Answered: Ronan 1341

March 23 2024

0 0

@dharr I appear to have the setup working in principal. I created seperate worksheets. One for the Main Package and one for the Sub Package. Both attached. I have hilighted the differences in both in red. Then I copied both help files across to the correct location. They now list correctly in Help.

New_RT_Edit_Dadabase_2024.mw

New_UHG_Edit_Dadabase_2024.mw

One way...

Answered: Ronan 1341

March 20 2024

1 2

Interesting problem. I found this method in the RealDomain help page.

restart;

n:=3;m:=2;
eqx:=x^(n/m)=a;
use RealDomain in (maple_sol:=solve(eqx,[x]))[] end use;  #also tried solve()
F:=map(X->eval(eqx,X),maple_sol);
map(X->evalb(X),F);
                             n := 3

                             m := 2

                               (3/2)    
                       eqx := x      = a

                          [     (2/3)]
                          [x = a     ]

                          F := [a = a]

                             [true]

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align and \n...

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Like this....

Doing it visually...

Suggestion:-One Drive or Google docs...

A proof...

One possible way...

How about using the Backup...

Use fsolve...

One way...

Check under Options Display...

It looks like I have it working correctl...

One way...

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