Maximum window

Geometry Level 3

Imagine a round window in the shape of a unit circle. It is meant to slide open as shown. The black dots must stay within the window frame.

What width of the sliding part will allow for the largest opening area when the window is fully opened, with the two dots hitting the circular window frame?

The answer is 1.414213562.

2 solutions

Jeremy Galvagni
Jul 23, 2018

The greatest area occurs when the horizontal and vertical lines form a square. The black diagonal has length $2$ and so the width of the window is $\sqrt{2}=\boxed{1.414213562}$

A nice proof without words

Roberto Gomide - 2 years, 10 months ago

I think the diagonal line is a bit missleading. One could think this was the solution...

Si Li - 2 years, 10 months ago

I thought a unit circle had a diameter of one so my answer was 0.707 ._.

Dominik Döner - 2 years, 10 months ago

A unit circle is definitely defined by its radius being one unit. The unit circle is also centered at the origin.

Jeremy Galvagni - 2 years, 10 months ago

Si Li
Jul 25, 2018

As you can see in the picture, the upper and lower boundary are just the opposite of each other, therefore you can replace them with straight lines. This leaves us with the problem of fitting an as large rectangle as possible inside a circle. Of course the rectangle must be a square. This circle is described by

x^2+y^2=1.

Since we fit in a square, x=y.

2x^2=1

x^2=1/2

x=+- 1/sqrt(2)

Therefore the width is 2* 1/sqrt(2)=sqrt(2)

Maximum window

The answer is 1.414213562.

2 solutions

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