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"Just Move It Over There, Dear!"

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"Just Move It Over There, Dear!"
The mathematics of moving mom's sofa 

by J. Schattman
Sir John A. Macdonald Secondary School
Waterloo, Ontario, Canada


My mother once asked me if I could please move her living room sofa into the guest bedroom down the hall and around the corner.  Before I broke my back dragging this battleship down the hallway only to discover that it wouldn't make the turn, I decided to take some measurements and work out the math first. 




Starting simple: Assume a widthless sofa! 


Let's start with the simpler problem by assuming the sofa has zero width.  The general case will follow easily. 


Idea:  Compute the shortest gap that the sofa has to pass through as it makes the turn.  That is, minimize Typesetting:-mrow(Typesetting:-mi( with respect to Typesetting:-mrow(Typesetting:-mi(If we're lucky, the sofa will be longer than this gap, and we won't have to move anything. 



Let h be the width of the hallway and Typesetting:-mrow(Typesetting:-mi( be the length of the line segment that just grazes the corner and touches both outer walls at an angle of  θ with the inner wall.  We first need to find the angle at whichTypesetting:-mrow(Typesetting:-mi( is minimum.  (Intuition demands that Typesetting:-mrow(Typesetting:-mi(, but rigor demands we prove it formally, which we do here.) 


We have Typesetting:-mrow(Typesetting:-mi( 

proc (theta) options operator, arrow; `+`(`/`(`*`(h), `*`(sin(theta))), `/`(`*`(h), `*`(cos(theta)))) end proc (1.1)


We solve Typesetting:-mrow(Typesetting:-mi( for Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( = `+`(`-`(`/`(`*`(h, `*`(cos(theta))), `*`(`^`(sin(theta), 2)))), `/`(`*`(h, `*`(sin(theta))), `*`(`^`(cos(theta), 2))))Typesetting:-mover(Typesetting:-mo(`/`(`*`(`+`(`*`(h, `*`(`^`(sin(theta), 3))), `-`(`*`(h, `*`(`^`(cos(theta), 3)))))), `*`(`^`(sin(theta), 2), `*`(`^`(cos(theta), 2))))Typesetting:-mover(Typesetting:-mo([[theta = `+`(`*`(`/`(1, 4), `*`(Pi)))], [theta = `+`(`-`(arctan(`+`(`/`(1, 2), `-`(`*`(`+`(`*`(`/`(1, 2), `*`(I))), `*`(`^`(3, `/`(1, 2)))))))))], [theta = `+`(`-`(arctan(`+`(`/`(1, 2), `*`(`*`(`/`(1... 



Typesetting:-mrow(Typesetting:-mi(As with any 3rd degree equation, this one has three solutions.  It looks like we want the real solution, Typesetting:-mrow(Typesetting:-mi(.   
Our intuition was right!


This means the smallest gap we have to squeeze through is the one formed when the sofa is at an angle of Typesetting:-mrow(Typesetting:-mi(with the wall.  How big is this gap in terms of h? 


Typesetting:-mrow(Typesetting:-mi( = `+`(`*`(2, `*`(h, `*`(`^`(2, `/`(1, 2))))))Typesetting:-mover(Typesetting:-mo(`+`(`*`(2.8284, `*`(h))) 


Punch Line:   

A zero-width sofa whose length is less than Typesetting:-mrow(Typesetting:-mn(will make the corner.
(We'll see how to account for the width of the sofa presently.)


Let's visualize this result when h = 1.5. 


Typesetting:-mrow(Typesetting:-mi( = `+`(`/`(`*`(h), `*`(sin(theta))), `/`(`*`(h), `*`(cos(theta))))Typesetting:-mover(Typesetting:-mo(`+`(`/`(`*`(1.5), `*`(sin(theta))), `/`(`*`(1.5), `*`(cos(theta))))Typesetting:-mo(Plot_2d 


Accounting for the sofa's width 


Now for the almost incredibly simple result:   


If the sofa's length plus double its width is less than Typesetting:-mrow(Typesetting:-mn(, the sofa will make the turn! 


Here a picture is worth Typesetting:-mrow(Typesetting:-msup(Typesetting:-mn(words. 




A quick rule of thumb when you don't have a calculator on you 


1.  Measure the width of the hallway (h) 


2.  Measure the length (L) and width (w) of the sofa. 


3.  If Typesetting:-mrow(Typesetting:-mi( is comfortably less than triple the width of the hall (a liberal estimate of Typesetting:-mrow(Typesetting:-mn(), you'll make it! 



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