I have an equation (parentheses are used because of VBA code)
Y=(P/(12E((bt^3)/12))*A
and i know every variables but not "b". Is there any way how to ask Wolfram Alpha to "redefine" (not solve) equation so I can see something like following: I tried to do it manually (but result is not OK)
b=((P/EY)*12A))/t^3
I wish to see how right equation will look.
Original equation is on picture below
where
equation in [,] I simplified by A
I'm not sure if there's a way to tell Wolfram|Alpha to rearrange for a particular variable; in general it will usually try to rearrange for x or y.
If I substitute b for x in your equation and use the following query:
solve Y - (P/(12E((xt^3)/12))*A) = 0
then Wolfram Alpha returns the result you're looking for: x (b) expressed in terms of the other variables. Specifically:
x = A P / (E t^3 Y) for tY != 0 and AP != 0
I know that your question was about Wolfram Alpha, that you do not want to "solve", but here is one way you could do it in Mathematica using your real question. I renamed I into J because I is a reserved symbol in Mathematica for the imaginary unit.
J = b t^3/12;
expr = (P / (12 E J) ) (4 L1^3 + 3 R ( 2 Pi L1^2 + Pi R^2 + 8 L1 R ) + 12 L2 (L1 + R)^2)
Solve[ Y == expr , b]
Result
{{b -> (P (4 L1^3 + 12 L1^2 L2 + 24 L1 L2 R + 6 L1^2 \[Pi] R + 24 L1 R^2 + 12 L2 R^2 + 3 \[Pi] R^3))/(E t^3 Y)}}
Related
I was looking at the problem and the discussion here: Easy interview question got harder: given numbers 1..100, find the missing number(s)
One of the user provided a solution using following equation.
k1 + k2 = x
k1^2 + k2^2 = y
Substituting provides (x-k2)^2 + k2^2 = y
I am trying to solve this equation further and come up with a C program to solve the problem of finding duplicates.
Inspite of spending lot of time I couldn't solve this equation to get k1 or k2 one side. I always ended up with k1 or k2 on both side of equation.
Any help is appreciated.
Expand the equation
(x - k2)^2 + k2^2 = y
and get
x^2 - 2xk2 + 2k2^2 = y
or
2k2^2 - 2xk2 + x^2 - y = 0
Now use the formula for solving the quadratic equation az^2 + bz + c = 0 which is (-b +/- sqrt(b^2 - 4ac))/2a. Only that in our case z=k2. So
k2 = (2x +/- sqrt(4x^2 - 8(x^2 - y))) / 4
or
k2 = (x +/- sqrt(x^2 - 2(x^2 - y))) / 2
= (x +/- sqrt(2y - x^2)) / 2
and you can put
k2 = (x + sqrt(2y - x^2)) / 2
k1 = (x - sqrt(2y - x^2)) / 2.
For a collision algorithm I am developing, I need to find out how to reflect a line over another.
Line 1:
y=ax+b
Line 2:
y=cx+d
Line 3:
(a result of reflecting line 1 over line 2) y=ex+f
Is there any algebraic way to determine e and f in terms of a, b, c, and d?
I have run over this exact same problem before. Stay with me here...
This problem involves two parts:
1. Find the point at which they intersect
to find where two lines intersect, we use the two equations of the lines:
y = M1x + B1
y = M2x + B2
Using substitution:
M1x + B1 = M2x + B2
M1x - M2x = B2 - B1
x(M1 - M2) = B2 - B1
x = (B2 - B1) / (M1 - M2)
To find the y value, just plug it in:
y = M1x + B1
2. Find the slope of the line from the other two slopes.
The second is far trickier. Using trigonometry, it is not impossible.
Let L1 be the "base line." (With a slope of M1)
Let L2 be the line that is to be reflected over the "base line." (With a slope of M2)
Let L3 be our resulting line. (With a slope of M3)
The equation I used is as follows:
double M3 = ((2 * M1) + (M2 * pow(M1, 2)) - M2) / (2 * M1 * M2 - pow(M1, 2) + 1);
Straight from my C code.
It is important to note that both slopes should be defined. You can use L'Hopital's rule to get an equation when one of the slopes is approaching infinity.
ONWARD WITH THE EXPLANATION!
Here is a crude drawing of three lines.
L2 is reflected over L1, resulting in L3. Drawing is not exact.
The angle between L1 and L2, as well as L2 and L3, is labelled as R.\
Here are the facts:
M1 = tan(A1)
M2 = tan(A2)
M3 = tan(A3)
This comes from the definition of tangent.
A3 = R + A1
This is a little trickier to see, but if you draw a horizontal line at the point of intersection it becomes obvious.
Thus, our goal is to find tan(A3). To accomplish this, we need to find R. As we can see, R can be found in a triangle with A2 and the supplement of A1 as the other angles. Thus, we know:
R + (180 - A1) + A2 = 180
R - A1 + A2 = 0
R = A1 - A2
Let's take the tangent of both sides:
tan(R) = tan(A1 - A2)
From trigonometry, we know:
tan(R) = (tan(A1) - tan(A2)) / (1 + tan(A1)tan(A2))
R = arctan((tan(A1) - tan(A2) / (1 + tan(A1)tan(A2))
Arctan being inverse tangent. From our earlier formula, A3 = R + A1, we get:
A3 = arctan((tan(A1) - tan(A2) / (1 + tan(A1)tan(A2)) + A1
A3 = arctan((M1 - M2) / (1 + M1*M2)) + A1
But we don't want A3. We want tan(A3). So again, we take the tangent of both sides.
tan(A3) = M3 = tan(arctan((M1 - M2) / (1 + M1*M2)) + A1)
M3 = tan(arctan((M1 - M2) / (1 + M1*M2))) + tan(A1) / (1 - tan(arctan((M1 - M2) / (1 + M1*M2))) * tan(A1))
Unfortunately, that's disgustingly hideous. Replacing tangents with slopes and simplifying, we get
M3 = ((M1 - M2) / (1 + M1*M2)) + M1 / (1 - ((M1 - M2)/(1 + M1*M2)) * M1)
M3 = (M1 - M2 + M1*(1 + M1*M2)) / (1 + M1*M2 - M1*M1 + M1*M2)
M3 = (M1^2 * M2 + 2*M1 - M2) / (1 + 2*M1*M2 - M1^2)
Which is the exact same as the formula above. Sorry about all the ugly math. When M2 is completely vertical, you can use L'Hopital's rule to get
M3 = (M1^2 - 1) / 2*M1
If anyone is so inclined, check my math. But I'm tired right about now.
Assuming the two lines are not parallel to each other
Step 1:
First find the intersection of the line y = ax + b with line y = cx + d , that is by solving comes out to be
m = (d - b) / (c - a)
Step 2:
The final line has point of the form (x , ex + f) , so wjat we know is the line joining the point and the corresponding image is perpendicular to the mirror line AND the the midpoint of the first point and its image lies on the mirror line. Solving for first requirement ....
(Slope of line joining point and its image) * (Slope of the mirror line) = -1
we get ...
c * (e*pt + f - a*n - b)/( pt - n ) = -1 -----> The first equation .
Then the midpoint of the point and its image lie on the central line , i.e.
Y coordinate of the midpoint - ( c* x coordinate of midpoint + d) = 0
y coordinate of midpoint = (a*n + e*pt) / 2 and x coordinate = ( pt + n) / 2
putting it above we get...
(a*n + e*pt)c - c( pt + n) - 2d =0 ----> second equation
3.
now the point and its image make equal angles from the intersection point .... a simple way of saying that angle between mirror line , point line and image line , point line being equal ... therefore ... the tangent of angle between lines mI and mM is equal to that of mM and mP
equating we get
( mM + mP ) / ( 1 + mp*mM) =( mI - mM )/ (1 + mI*mM)
where mM = c , mI = e, and mP = a -----> third equation
put it in their respective
places you get three equations in three unknowns , pt , e and f and solve ... just x in place of n earlier and there you have your e , f in terms of a , b , c , d.
Solve it yourselves ....
However if they are parallel its simple , you have two equations in two variables use the midpoint method
Yet another method:
Matrix of affine transformation for reflection relatively to line y=ax+b (works for non-vertical lines!).
Let's pa = 1+a^2, ma = 1-a^2, then matrix is (from Nikulin's Computer Geometry book)
ma/pa 2a/pa 0
2a/pa -ma/pa 0
-2ab/pa 2b/pa 1
So we can get two arbitrary points at second line, apply this transform, and calculate new line equation
I have the following expression
(-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B))
How can I multiply both the denominator and numberator by p^(A+B), i.e. to get rid of the denominators in both numerator and denominator? I tried varous Expand, Factor, Simplify etc. but none of them worked.
Thanks!
I must say I did not understand the original question. However, while trying to understand the intriguing solution given by belisarius I came up with the following:
expr = (-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B));
Together#(PowerExpand#FunctionExpand#Numerator#expr/
PowerExpand#FunctionExpand#Denominator#expr)
Output (as given by belisarius):
Alternatively:
PowerExpand#FunctionExpand#Numerator#expr/PowerExpand#
FunctionExpand#Denominator#expr
gives
or
FunctionExpand#Numerator#expr/FunctionExpand#Denominator#expr
Thanks to belisarius for another nice lesson in the power of Mma.
If I understand you question, you may teach Mma some algebra:
r = {(k__ + Power[a_, b_]) Power[c_, b_] -> (k Power[c, b] + Power[a c, b]),
p_^(a_ + b_) q_^a_ -> p^b ( q p)^(a),
(a_ + b_) c_ -> (a c + b c)
}
and then define
s1 = ((-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B)))
f[a_, c_] := (Numerator[a ] c //. r)/(Denominator[a ] c //. r)
So that
f[s1, p^(A + B)]
is
((1 - p)^B*p^A)/((1 - p)^(A + B) - p^(A + B))
Simplify should work, but in your case it doesn't make sense to multiply numerator and denominator by p^(A+B), it doesn't cancel denominators
I am using Mathematica 7 in the notebook interface and I want to rearrange an inequality so that I get a certain variable on one side. For eg.
FullSimplify[x^3+L+r>3x^3+2r]
Gives
L > r + 2 x^3
However, I want :
r < L-2x^3
Is there anyway we can instruct FullSimplify to order variables in a particular way? I am using Mathematica for presentation as well so, the way I arrange the variables is important to me.
Thanks
SR
Edit: I tried Reduce, while that works for this example, it does not work for the actual expression I have, I get an error saying,
This system cannot be solved with the methods available to Reduce.
Edit: here is the actual expression:
{L - (m^2 ((-2 + e)^2 \[Delta] + (5 +
2 e (-7 + 4 e)) \[Tau]) \[Omega])/(36 (2 - 3 e + e^2)^2)} > {0}
I want this to be displayed in the form of \[delta]< *something*
Thanks!
First of all, getting Mathematica to output something exactly as you would like it is something of a black art, and requires a lot of patience. That said, if you apply Reduce to your original expression, as per Belisarius, you'd get
In[1]:=Reduce[x^3 + L + r > 3 x^3 + 2 r, r, Reals]
Out[1]:= r < L - 2 x^3
However, as you pointed out, this isn't the full expression, and Reduce produces what can only be described as a less than helpful answer when applied to it. It is at this point where patience and a lot of extra processing is required. I'd start with
In[2]:=Reduce[ <full expression>, Delta, Reals] // LogicalExpand // Simplify
While this doesn't give you a clean answer, it is better than before and reveals more of the structure of your solution. (I would not use FullSimplify as that mixes Delta in with the other terms.) At this point, we need to know more about the terms themselves, and the output from In[2] is not quite as useful as we want.
I'd re-expand this with LogicalExpand which gives you twelve terms that are significantly simpler than the what Reduce alone gives. (You'll note that only the last six terms actually involve Delta, so I'd check that the variable conditions actually match those.) Selecting those last six terms only,
In[3]:=%2[[-6;;]] // Simplify
Out[3]:= m != 0
&& ((Omega > 0 && Delta < something) || (Omega > 0 && Delta < something else)
&& (1 < e < 2 || e < 1 || e > 2)
The third term is tautological, but Simplify nor FullSimplify can't seem to remove it. And we're really only interested in the middle term anyway. If Omega > 0 your expression can then be extracted via %[[2,1,2]].
Putting this all together in one expression:
In[4]:=Simplify[LogicalExpand[Reduce[<expression>, Delta, Reals]]][[-6;;]] //
Simplify // #[[2,1,2]]&
Out[4]:= Delta < something
After writing that out, I realized that there is a much simpler way to approach this. I'd redo line 2, above, as follows:
In[5]:= Reduce[ <full expression>, Delta, Reals] // LogicalExpand // Simplify //
Cases[#, ___ && Delta < _ && ___, Infinity]&
Out[5]:= {Omega > 0 && Delta < something}
Or, provided you really do know that m != 0 and Omega > 0 you can do
In[6]:= Reduce[ <expr> && m!=0 && Omega > 0, Delta, Reals ] // LogicalExpand //
Simplify // #[[2]]&
Reduce[x^3 + L + r > 3 x^3 + 2 r, r, Reals]
Will do.
As I don't use Mathematica for editing or presentation, perhaps someone else may come with some extra advice.
Edit
based on your comment, you may try:
Reduce[{L - (m^2 ((-2 + e)^2 Delta + (5 +
2 e (-7 + 4 e)) Tau) Omega)/(36 (2 - 3 e + e^2)^2) > 0}, Delta, Reals]
Where I corrected some syntax errors. But you'll find that the resulting expression is rather unpleasant. To simplify it further you need to know the valid ranges for your vars. Please post that info if you have it.
HTH!
Inspect the output of
r=Simplify[Reduce[L-(m^2((-2+e)^2\\[Delta]+(5+2e(-7+4e))\\[Tau])\\[Omega])/(36(2-3e+e^2)^2)>0,\\[Delta],Reals]]
to see that
r[[2,1,1,1]] gives \\[Delta]>expr,
but
r[[2, 1, 2, 2]] gives \\[Delta]< expr,
because the sign of \[Omega] in the denominator of expr. All this ignores the other conditions on the values of L, e, m and \[Omega] that will change the result and different versions of Mathematica may change the form of the result from Simplify[Reduce[]] which will invalidate all of this.
Part of the difficulty in reducing the expressions returned by Reduce[] and LogicalExpand[] is that the supplied expression involves division by zero when e=1 or =2.
I get something bearably compact with
Assuming[{
(L | m | e | Tau | Omega | Delta) \[Element] Reals
},
FullSimplify[
LogicalExpand[
Reduce[{L - (m^2 ((-2 + e)^2 Delta + (5 +
2 e (-7 + 4 e)) Tau) Omega)/(36 (2 - 3 e + e^2)^2) >
0}, Delta, Reals]
]
]
]
Out[]:= (L > 0 && (1 < e < 2 || e < 1 || e > 2) && (m == 0 || Omega == 0)) ||
(m != 0 && (
(Omega > 0 &&
Delta < (36 (-1 + e)^2 L)/(m^2 Omega) + ((-5 + 2 (7 - 4 e) e) Tau)/(-2 + e)^2) ||
(Delta > (36 (-1 + e)^2 L)/(m^2 Omega) + ((-5 + 2 (7 - 4 e) e) Tau)/(-2 + e)^2 &&
Omega < 0)) &&
(e > 2 || e < 1 || 1 < e < 2))
where I've expended no effort to replace symbol names with symbols.
(Why Assuming[...]? Because I'm too lazy to remember to get the same assumptions jammed into each simplification step.)
I have this problem containing some inequations and requirement to minimize a value. After doing some research on the Internet, I came to conclusion that using Prolog might be the easiest way to solve it. However, I never used Prolog before, and I would hate to waste my time learning it just to discover that it is not the right tool for this job.
Please, if you know Prolog, take a look at this problem and tell me if Prolog is the right one. Or, if you know of some other language that is really suited for this.
a + b + c >= 100
d + e + f >= 50
g + h >= 30
if (8b + 2e + 7h > 620) then y = 0.8 else y = 1.0
if (d > 35) then x = 0.9 else x = 1.0
5xa + 8yb + 5c + 3xd + 2ye + 2f + 6xg + 7yh = w.
I need to find the values for a, b, c, d, e, f, g and h that minimize w.
Please note that the above is only an example. In real program, I would use up to 10000 variables and up to 20 if..then clauses. This rules out linear programming as an alternative technique because it would take a prohibitive amount of RAM and time to test all LP problems.
I'm not really asking for code, although I'd be grateful for some hint how to tackle this if Prolog is really good for it. Thanks.
You could have a look at Constraint Logic Programming, either CLP(R), CLP(Q) or CLP(FD).
Your problem can be encoded into CLP(R) as follows (I think):
:- use_module(library(clpr)).
test(sol([A,B,C,D,E,F,G,H], [X,Y,W])) :-
{A >=0, B >=0, C>=0, D>=0, E>=0, F>=0, G>=0, H>=0},
{A + B + C >= 100},
{D + E + F >= 50},
{G + H >= 30},
{5*X*A + 8*Y*B + 5*C + 3*X*D + 2*Y*E + 2*F + 6*X*G + 7*Y*H = W},
(({8*B + 2*E + 7*H > 620},{Y=0.8}) ; ({8*B + 2*E + 7*H =35},{X=0.9}) ; ({D=
Using SICStus Prolog, I get the following answer:
| ?- test(A).
A = sol([_A,0.0,_B,0.0,_C,_D,30.0,0.0],[1.0,1.0,780.0]),
{_A=100.0-_B},
{_C=50.0-_D},
{_B==0.0},
{_D>=0.0} ? ;
no
You can solve this using linear programming (LP), but the problem needs some modification before you can chuck it into an LP solver. An LP problem basically involves maximising or minimising a function given some constraints.
First, split the problem into two problems (as LP does not support the two if conditions you have):
Constraints:
a + b + c >= 100
d + e + f >= 50
g + h >= 30
8b + 2e + 7h > 620
Linear function:
5 * 0.8a + 8 * 1.0b + 5c + 3 * 0.8d + 2 * 1.0e + 2f + 6 * 0.8g + 7 * 1.0h = w
And
Constraints:
a + b + c >= 100
d + e + f >= 50
g + h >= 30
d > 35
Linear function:
5 * 1.0a + 8 * 0.9b + 5c + 3 * 1.0d + 2 * 0.9e + 2f + 6 * 1.0g + 7 * 0.9h = w
After you run both separately by the LP solver, the solution will come out with the values of a, b, c, d, e, f, g, h and w. Pick the smaller value of w and the corresponding values of a, b, c, d, e, f, g, h.
How does this work?
The two if conditions are effectively similar to the other constraints listed, just that they entail different values of x and y. Since the two conditions are mutually exclusive (given that both cannot be satisfied as x and y will hence have different values), you can split them into two separate LP problems. As a result, you solve the LP problems individually, and hence you will arrive at a minimised value of w.
For an LP solver, go to the linear programming Wikipedia article as linked above. There are tools like Excel and some others which are easier to use, but if you want a program, then there are numerical languages which are good at this, or general purpose languages like C that can do this with a library like glpk.
Hope this helps!
I haven't worked with similar problems before, so I can't give you any direct suggestions. However, I would not use Prolog for it. Prolog is really good for dealing with symbolic problems (a classic example would be the Einstein puzzle), but its math support is very awkward; it feels like math was tacked on as an afterthought.