# How to determine the location of a mark on an object that has changed size?

I apologize up front for the horrible title, I do not have the mathematics vocabulary to eloquently summarize this in a title.

This first picture and question is a lead-up to the actual question. In an attempt to explain it.

You have an object of a known size, you put a mark on it at a certain percentage of it's size. You now increase the size of that object in one specific direction, you know by what percentage that object changed size, at what percentage is that mark currently located?

That's a fairly simple answer. Now knowing the previous question, consider the following:

You have an object of an unknown height. You have a mark that has a known height as a percentage of the height of the object it takes up and a known location on that object as a percentage of the height of that object. You increase the height of the object by an unknown amount. The only thing you now know is the height of the mark as a percentage of the height of the object. How do you find it's location as a percentage of the height of the object?

Edit: The green object ONLY expands down, and the red mark stays in the exact same position relative to the top o the object. The red mark also maintains the exact same size in units, even though that size is unknown.

I hope that made sense. If that is unsolvable, what other variable do I need? (Additionally, what are appropriate tags for this question?)

Edit: Oddly enough you can intuitively solve this when the numbers are simple. If the red mark is at 50% of the height and the green box increases in size downwards by 100% of it's current size. Then the red mark will be at 75% of it;s height. I'm not sure if this is an intuitive guess that just ends up being correct, or if there is some sort of mathematical backing.

I assume your mark doesn't change position relative to the upper bound of the expanded object (looks so to me based on your picture).

Let $x$ be the height of the mark, $y$ be the height of the original object, $z$ be the height of the expanded object. We have

$$\frac{x}{y}=50\%=\frac{1}{2}\\ \frac{x}{z}=0.27$$

Divide these two equations:

$$z=\frac{y}{0.54}$$

Now let $y$ be $1$. The center of the mark is at $1-0.65=0.35$. Suppose it does not change, but the height of the object is now $\frac{1}{0.54}$. So the center of the mark is now at

$$1-\frac{0.35}{\frac{1}{0.54}}=1-0.35\cdot 0.54=0.811$$

• Thank you! I'm just writing this down into an equation I can plug values into right now. I am stuck on where you got 0.35, where did that number come from? – Douglas Gaskell May 12 '15 at 19:12
• @douglasg14b: Sorry forgot to mention that. I edited the answer. – KittyL May 12 '15 at 19:18
• This is probably going to sound ridiculous, but now where did the 0.65 come from? If z = y/0.54 (which is the same as z = y* 1.851) this determines the change in size, z is 1.851 times larger than y. Maybe I'm a bit dull, though I'm not quite getting where the 0.65 comes from if y = 1; – Douglas Gaskell May 12 '15 at 19:24
• @douglasg14b: It is because you said the center is at $65\%$ of the height. – KittyL May 12 '15 at 19:26
• Ohhh, I see. Thanks, I was being pretty dull there. – Douglas Gaskell May 12 '15 at 19:28

It depends how the object is expanded.

If it is expanded uniformly by a certain factor then the size of the mark is also expanded by the same factor and its location is unchanged.

If the asparagus analogy is correct, then the length below the mark is fixed and only the length above the mark actually grows. If the object expands by a factor of $z$ (let's take the initial length of the object as our unit length) and our mark's mid-point is at $x$ (with $0$ being the bottom of the object) then the position of the mark after expansion is at $x/z$. If the mark's position is with reference to the top of the object (i.e. $0$ means the midpoint of the mark is at the top of the object) then, its new location is at $1-(1-x)/z$.

• Thank you for the reply. The marks size in units is always constant (however it's actual size is unknown), the only object that changes in actual size is the green box. Which will always be expanded downwards. – Douglas Gaskell May 12 '15 at 18:07
• As a visual analogy, could the object be an asparagus (growing upwards) and the mark an independent plastic label fixed to the ground? – dumb0 May 12 '15 at 18:18