Find the maximum area of rectangle R so that it will still be

>>8236798
>maximum area that can go through the corner
Just push it to the right, then push it upward. 2 meters high by 4 meters wide. 8 square meters.

unsolved
https://en.wikipedia.org/wiki/Moving_sofa_problem

I got
[eqn] R = (8 \sqrt{5} - 8) \; \text{m}^2 \approx 9.889 \; \text{m}^2 [/eqn]

>>8236846
The more general problem is unsolved but this question specifies that the sofa should be rectangular.

File: Hammersley_sofa_animated.gif (265KB, 280x200px) Image search: [Google]

265KB, 280x200px

>>8236879
The wikipedia article states that this is not the largest solution (with certainty, somehow).

I can't see how it isn't the largest solution.

>>8236918
The more general problem is unsolved but this question specifies that the sofa should be rectangular.

>>8236918
The question being asked now isn't the largest shape, it's the largest rectangle. Hammersley's sofa isn't a rectangle.

>>8236918
did you not see the sentence right after? it says
>Joseph Gerver found a sofa that further increased the lower bound for the sofa constant to approximately 2.2195

>>8236925
>>8236930
Did I say I was talking about OP's question? If I wanted to address OP, I wouldn't have replied to the post I replied to.

>>8236814
But can you prove that this is the largest :)

>>8236871
How did you get this? I'm clueless

>>8236798
What is limits of angles between sofa and wall???

>tfw this was in my calc 1 final
Am I a brainlet for not knowing how?

>>8237173
about tree fiddy

Im on my phone so the photo may not come be straight, but this is what I got.

File: 20160730_171606.jpg (1MB, 3264x2448px) Image search: [Google]

1MB, 3264x2448px

>>8237280

>>8237284
Oops and on one part I meant l<2 not l>2

>>8237217
no
this problem is unsolved

>>8237284
>yet still can't take a picture the right way around

>>8237506
It wouldn't fit round the bend.

File: bueno69.png (410KB, 500x514px) Image search: [Google]

410KB, 500x514px

>>8237542

>>8236798
Max area is approximately 8.8 m2

>>8237506
Uploading from android phones using camera usually turns the pic sideways

>>8236798
For a rectangle of width W and length L that touches hallways of width A and B when placed on an angle of θ with respect to the hallway, such that W<A<B, we get:

L = Acsc(θ) + Bsec(θ) - Wcsc(θ)sec(θ)

For fixed values of W, A and B, the maximum length of the rectangle that fits around the corner corresponds to the minimum value of L(θ).

dL/dθ = 0

Solving the differential equation gives:
Wcos(2θ) = Acos3(θ) - Bsin3(θ)

Let W = 4/√5, A=2, B=4:
Solving gives:
θ = -2arctan(2-√5)

Substituting into the length formula above:
L = 4.472135

This yields the maximum area of the rectangle as:
LW = 8.000000

The result i got is: Area max = 8.0
In fact:

i used desmos.com to solve the final system,
to see how the value of the function in x=4 changes while changing a and b, and it shows that the maximum value of the product a*b that respects the inequalities(0<b<6, 0<a<2, f(4)<2) was a*b=8.. desu i tried solving it using derivatives but it was an horrible derivative so i gave up and i used another tool to find the solution..
Ps:with ∀θ i mean ∀θ, 0<θ<pi/2 >_>

>>8237800
>>8238507
>>8238508
>>8238518

I used similar methods and got 8 as well.

>>8236918
I'm in engineering so I'm retarded

I've never understood how mathematicians can like prove that a proof exists but not actually prove it. Is this just bad headline writing that I've been reading?

>>8239211
Yes. A proof that a proof exists is a proof.

What you may have read is that mathematicians can prove that there is neither a proof nor a disproof. Logicians do this regularly.

>>8239259
I guess my question is how exactly do they prove a proof exists without actually doing it

I should probably just Google this

>>8239417
First, you still misunderstand. A proof that a proof of P exists *is* a proof of P.

But to address what may be your question, on a more circumlocuitous level one can use Gödel's completeness theorem to show that a formal proof exists by showing there is no model that satisfies the negation.

Metamathematics.

>>8236798
Just flip the couch up on its end, silly.

>>8239417
It is just special wording and does not mean the same thing in laymen's use of the term.

>>8236984
In order to increase the area of the rectangle, the length of one or both sides must be increased. We cannot increase the length of either side because we are constrained by the width of each hallway. Therefore 8 is the max. QED

>>8239537
It's a house for manlets though and the ceiling is only 2m high.

>>8237697
Take the pic at an angle that anticipates a 90 degree turn