The Ladder around the Corner Problem

test

move the

ladder

around the

corner

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

maximum length of ladder: 12.5

This Demonstration visualizes a typical problem of constrained optimization: find the maximum length of a ladder that can be moved horizontally around the corner between two perpendicular corridors of different widths

a

and

b

.

The ladder touches the outer walls at the points

(x,0)

and

(0,y)

, and the corner at

(a,b)

. In order for the ladder not to intersect the gray area,

y(x)≤bx/(x-a)

throughout the ladder's range of motion. That is, the red point must stay below the hyperbola

y(x)=bx/(x-a)

(orange). Since the ladder has fixed length

ℓ

, the red point must be on the circle

ℓ(x)=

x

2

+y(x)

2

(blue).

The blue circle touching the orange hyperbola at the green point represents the largest such circle with radius

ℓ

max

. The coordinates of the green point are

(x,y)=a+a

1/3

b

2/3

,a

2/3

b

1/3

+b

. The length of the ladder is then

ℓ

max

=

a

2/3

+b

2/3



3

[1].

The Locator defines the corner at

(a,b)

. The slider lets you move the ladder around the corner to verify the solution visually. The "test" button sets the ladder to its maximum length at its critical point.