Question 724109
1 inch=2.54 cm (centimeters) (It's not 2.52).
Your logo (a design drawn on a piece of paper) has been chosen and its measurements (in cm) must be used in the calculations.
(Nowadays, they probably move from rough paper sketches to computer design, and measure sizes in pixels, but let's pretend we draw on paper and use calculators).
 
Let's say an enlarged logo design is presented on an 8.5" by 11" piece of paper, looking like this:
{{{drawing(390,300,-6,7,-5,5,
rectangle(-5.5,-4.25,5.5,4.25),
green(rectangle(-3.5,-3,3.5,3)),
locate(5.6,0.5,8.5),locate(5.6,0,inches),
locate(-1,-4.25,11),locate(-0.2,-4.25,inches),
locate(-0.1,-3.1,W),locate(3.7,0.2,H),
arrow(-0.5,-3.3,-3.5,-3.3),arrow(0.5,-3.3,3.5,-3.3),
arrow(3.8,-0.5,3.8,-3),arrow(3.8,0.5,3.8,3),
locate(-2.5,2,logo),locate(1,0.5,fits),locate(-1.5,-1.5,here)
)}}} The logo fits in the green rectangle (touches each of the edges at some point).
The green rectangle is {{{W}}} cm wide and {{{H}}} cm tall.
On the billboard it will be scaled up by a factor {{{x}}} to be {{{Wx}}} cm wide and {{{Hx}}} cm tall.
 
Let's say that you want to put that logo on a billboard that is {{{12feet}}} tall and {{{24feet}}} wide. That is
{{{(24feet)(12inches/foot)(2.54cm/inch)=731.52cm}}} wide and
{{{(12feet)(12inches/foot)(2.54cm/inch)=365.76cm}}} tall.
Do you want to leave some space around the edges of the billboard?
Or would it be OK to make your logo {{{12feet}}} tall?
Let's say it is OK if on the billboard the logo is {{{12feet}}} tall, and it would also be OK if it is {{{24feet}}} wide.
Obviously, you cannot make it larger, because you would have to leave pieces of the logo out of the billboard.
So, the limits for the size of the logo (in cm) are:
{{{Wx<=731.52}}} --> {{{x<=731.52/W}}}
and {{{Hx<=365.76}}} --> {{{x<=365.76/H}}}
You take the samllest of {{{731.52/W}}} and {{{365.76/H}}} as your maximum scale factor.
Once you find {{{W}}} and {{{H}}} in centimeters (the size of your paper logo), you can find the maximum values of the scale factor {{{x}}} to make sure your
logo is not too wide or too tall for the billboard.
 
IF I had a logo that fits in a rectangle 18cm wide and 15cm tall, I would calculate
{{{x<=731.52/18=40.64}}} to make sure it's not too wide
and {{{x<=365.76/15=24.384}}} to make sure it's not too tall.
So I may decide to use a scale factor of 24.384 to make the logo as tall as the billboard. It will not be as wide as the billboard, but I will figure out what to do with the part of the billboard that is not filled up with the logo. Maybe I'll fit the logo into one side of the billboard and use the rest for some short message in text like "Best in the state." Or maybe I'll center the logo and just add background on both sides.