Fitting 1 cm^2 rectangle with the highest possible aspect ratio inside a 20mm x 15mm rectangle

[LaraKraft]

I have a problem that seemed easy at first but I couldn't manage to solve it. I have a rectangle with 20 mm x 15 mm dimensions. I want to fit a 1 cm^2 rectangle inside the larger rectangle with the highest possible aspect ratio (Length/Width). I would be happy if someone could help me with this problem. A basic schematic of the problem is attached.

 

[khansaheb]

I have a problem that seemed easy at first but I couldn't manage to solve it. I have a rectangle with 20 mm x 15 mm dimensions. I want to fit a 1 cm^2 rectangle inside the larger rectangle with the highest possible aspect ratio (Length/Width). I would be happy if someone could help me with this problem. A basic schematic of the problem is attached.View attachment 37752

Area of your LARGER rectangle = 15 * 20 mm² = 300 mm²

Area of your SMALLER rectangle = 1 cm² =100 mm²

Assume that

the length of the small rectangle is = l * 10 mm​

​

the width of the small rectangle is = 1/l * 10 mm​

Continue.....

 

[blamocur]

I want to fit a 1 cm^2 rectangle inside the larger rectangle with the highest possible aspect ratio (Length/Width).

Does the smaller rectangle have to be inscribed in the larger one ? I.e., should all vertices of the small rectangle lie on the sides of the larger one.
Also note that in your drawing the smaller polygon looks like non-rectangular parallelogram.

 

[Dr.Peterson]

I have a problem that seemed easy at first but I couldn't manage to solve it. I have a rectangle with 20 mm x 15 mm dimensions. I want to fit a 1 cm^2 rectangle inside the larger rectangle with the highest possible aspect ratio (Length/Width). I would be happy if someone could help me with this problem. A basic schematic of the problem is attached.View attachment 37752

Does the smaller rectangle have to be inscribed in the larger one ? I.e., should all vertices of the small rectangle lie on the sides of the larger one.
Also note that in your drawing the smaller polygon looks like non-rectangular parallelogram.

I suspect that the requirement of maximal aspect ratio may conflict with wanting all four corners to touch; either of the two by itself may determine a single solution. (I'm not sure whether the latter is necessarily implied by "inscribed", though it's what we usually think of; but in any case, the word wasn't used here.)

It may help if you tell us the reason for the question. Is this for some practical reason, or is it, say, homework or a contest problem?