ok jugoslav,
I will mark just one of your errors
DB = sqrt(2 * R) sqrt(2 * R2) wrong
\(DB=\sqrt{R^2+R^2}\\ DB=\sqrt{2R^2}\\ DB=\sqrt2 * \sqrt{R^2}\\ DB=\sqrt{2} *R\)
Your answer does not give a ratio at all. We already know that the ratio is r:R that is given in the question.
Yours does not simplify to that so it is not correct.
PLUS the whole point is to put R in terms of r
Note: it is really good that you are questioning my solution.
That shows everyone here that you are actively learning.