I have a twine 5 meters long. I would like to arrange the word "MARYSIA" from that tape, keeping all letters the same size, it is ok to cut the twine. I want to choose well known Arial typeface, so I could easily determine the ratios of the letters. How I could determine the size of the letters? First, it could be height, then all other dimensions from the ratios. It could also be an iterative way.
When I am thinking about this problem, I would start with one letter. I would the twine on the floor and would form the letter M (it would be big, 5 meters is a lot of the twine). Then I could shrink the first letter "M" and take the spare twine to form "A", and so on. Finally, I would get all letters in the same size.
Please, how to proceed with such a task only theoretically? Just using pencil and paper (and finally a computer, I can do some calculations using JavaScript)
Start from the fact that the length of each string needed for a specific letter scales linearly with its height $h$ and also linearly with its width $w$. Keeping the aspect ration this means $$ l_i(\gamma h,\gamma w) = \gamma^2 l_i(h,w) $$ Doubling the size means you need four times the length. You can think of starting with fixed $h$ and $w$ and adjusting $\gamma$ until $$ \sum_i l_i = 5m $$ From this you can calculate $\gamma$ $$ 5m = \sum_i l_i(\gamma h,\gamma w) = \gamma^2 \sum_i l_i(h,w) $$ since you know the initial length $\sum_i l_i(h,w)$. This leads to $$ \tilde{l}_i = l_i(\gamma h,\gamma w) = \frac{5m}{\sum_i l_i} l_i $$ where the $l_i$'s are the initial lengths of letter $i$ and the $\tilde{l}_i$'s are the letters you want to use.