simito_ said:
While calculating the surface area for an object, I was told the below statement. However, I am not sure is this correct, please can someone help me to explain this with an example? Is the below statement always true?
The surface area % increase should be in line or less than the % length increase.
Yes, it is a correct statement. [Edit: Providing that we are scaling only "length" and not "width" and "height" as well].
For a flat object such as a sheet of paper, you can see that the surface area will be directly proportional to the object's length, right?
For an object with some extent in the third dimension, things are trickier. You can see that the cross-sectional areas of the top and bottom will have increased. The cross-sectional areas of the front and back (looking perpendicular to the length axis) will have increased as well. But the cross-sectional areas of the two ends will have remained unchanged. This is admittedly a bit hand-wavy because the surface area of a shape is not equal to the sum of the cross-sectional areas of its six sides.There is a more rigorous way to prove the inequality. I am not sure that I can do justice to the reasoning in a short post...
Suppose that we imagine the surface of the unstretched object tiled with a bunch of rectangular panes. Maybe you fill in the gaps with strange shapes. But it is clear that you can tile essentially the whole shape with rectangular panes if you make them small enough -- you just use smaller and smaller panes as you fill in whatever irregular gaps remain. [We are doing a surface integral].
We make sure that all of the panes are lined up with a "width" axis that is perpendicular to the direction of stretch. The "length" axis for each pane will be more or less parallel to the direction of stretch.
If it helps, imagine a spherical Earth being stretched by pulling the north and south poles apart. We divide the surface into north-south strips. The strips would be wedge shaped, so we trim to make them rectangular and cover the places where we trimmed with smaller rectangles, repeating the process indefinitely.
When we stretch the object, the width (east/west) of each of the panes will remain unchanged.
When we stretch the object, the length (more or less north/south) of each of these panes will increase by
at most the percentage of stretch. Near the poles, there will be negligible increase. Near the equator there will be the full increase.
So the surface area of each pane will increase by at most the percentage of stretch. So the total surface area of the object will increase by at most the percentage of stretch.