Calculating aspect ratio limits of rotated rectangle within a rectangle

Question

Hello !

I was wondering if there would be an equation for this sort of a problem or if the solution might need to be brute forced by approximation.

example image: https://imgur.com/a/NI6DJli

There is a rectangle A of a known size in a 4:3 aspect ratio.Centered inside of A there is another rectangle B also of 4:3 aspect ratio, which is of variable size and variable rotation. In my example IMG B is 40x30 and rotated by 30 degrees.I would like to calculate by how much I can make the aspect ratio of B "wider" so that it would retain the areaa size and touch the "boundaries" set by A.

I know how to calculate the distance from the corner of B to A, if following the line of B. However whenever I change the ratio the distance changes. And that left me a bit bewildered on how to go about what is the relationship between the ratio and the distance.

Thank you very much for any possible thoughts !

Answer 1

Hi Alex s.,

Wyzant just wiped out my answer, I'll try again.

1) Draw your desired target figure.

2) Label inner rectangle sides as x (the more vertical edge) and 1200/x (the more horizontal edge)

3) label segments of the outer rectangle from lower right corner CCL respectively as p1, p2, p3, p4.

4) Use trig and the 30 degree triangles set up to express p1 ... p4 in terms of trig(30) and x variously. You should get 600/x, rt(3)x/2, x/2, and 600rt(3)/x respectively?

5) p1 + p2 = (3/4)(p3+p4)

6) Substitute, collect terms, solve for x

I suspect 75 and 16 are bogus.... but you will see.

-- Cheers, --Mr. d.