Brain work, geometry edition

[Ptah]

With a square 48 inches, what is the largest diameter of two circles that will fit inside?

Edit to add: the circles are the same size.

[yonder]

1. 48 inches. A large one of 48" and a much smaller in one of the corners.

That answers the question accurately I think. If the circles are to be the same size as shown, I'm thinking.

[Ptah]

2. Yes, I meant to say they are the same size.

[yonder]

3. Sorry for the smart ass answer....just testing for a trick question.

[Ptah]

4. It was a clever answer.

[RockRaven]

5. If it is two identical circles, then

I think the diameter would be equal to 48/(1+1/√2)

[ProfessorGAC]

14. That's How I Did It

I get 28.12"

[Ptah]

17. Well done.

[Ptah]

16. Impressive! You solved it in less than 30 minutes'

[gratuitous]

6. If I remember my Pythagorean Theorem

A square of 48 inches per side would yield a diagonal of 67.88 inches, which would accommodate two circles of diameter 33.94 inches.

[Ptah]

8. That would put much of the circles outside of the square.

[intrepidity]

7. 33.94? nt

[Ptah]

9. That would put much of the circles outside of the square.

[intrepidity]

12. Oops yeah I see my mistake now doh nt

[yonder]

10. 33.94 is too large as it creates a diameter from the midpoint of the hypotenuse

to the intersection of the tangents. The diameter is going to be smaller in the neighborhood of 28 and change but I’m wrestling with how to calc it.

Edit: what the OP just said.

[Ptah]

18. Correct, 28.1.

[intrepidity]

19. So....what's the equation, exactly? nt

[Ptah]

20. RockRaven correctly wrote it like this:

RockRaven (12,587 posts)
5. If it is two identical circles, then

I think the diameter would be equal to 48/(1+1/√2)

https://www.democraticunderground.com/?com=view_post&forum=1018&pid=1803704

[intrepidity]

21. I mean, using the variable R and the known 48. nt

[Ptah]

22. Sure:

48/((2*R)+(2*R)/√2)

[intrepidity]

23. I must be dense

If we call the dashed line "s" then

2R + s = 48
or
s = 48 - 2R

We know that, for hypoteneus h:

h^2 = 2s^2
and
h = 2R
so
4R^2 = 2s^2

Since
s = 48 - 2R
then
4R^2 = 2(48-2R)^2

Ah, ok, now I see.

[intrepidity]

13. 27?

Without using formulas, so probably wrong.

Logic is: 24 is too small, 30 is too big, so split the difference.

[Jack the Greater]

15. Beyond my ability

RockRaven's answer seems convincing, but I am not familiar with the math to get to it. Square root of 2 was never my forte