SOLUTION: 4 semicircle and 4 quadrant each of radius 7cm were cut off from a rectangular cardboard if perimeter 128cm.Find the perimeter of the cut cardboard.

Question 1073352: 4 semicircle and 4 quadrant each of radius 7cm were cut off from a rectangular cardboard if perimeter 128cm.Find the perimeter of the cut cardboard.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Without more detail, some assumptions need to be made.

1) For starters, we need a conveniently shaped rectangle,
with enough usable surface area, such as a 16 cm by 48 cm rectangle.
Not all rectangles with a perimeter of 128 cm will yield
4 semicircles and 4 quadrants each of radius 7cm.
For example, a rectangle of with 8 cm and length 56 cm has a perimeter of
,
but it cannot yield 4 semicircles and 4 quadrants each of radius 7cm,
not even reshaping the cutouts to make circular sectors out of the wasted cutout pieces.
That rectangle has an area of
,
while 4 semicircles and 4 quadrants each of radius 7cm
add to a larger total surface area.
They have the same total area as the 3 circles of radius 7cm they would form,
.

2) Then, there is the issue of where to cut from.
One practical way, would be to cut the quadrants from the corners,
and the semicircles along the long edges of the rectangle,
with the straight sector edges along the sides of the rectangle,
like this:

With the assumptions above about
1) the shape of the rectangle, and
2) the way the circular sectors were cut off,
we are left with a piece that has for perimeter
some segments of the rectangle's straight edges, and
the curved edges of 3 circles.
The curved edges of 3 circles add up to

For each circular sector piece cut off,
the removed segment(s) of the rectangle's edges were

So what's left as straight segments of the rectangle's original perimeter is

So the straight plus curved parts of the perimeter of the
piece leftoever of the rectangle is approximately
.
If an exact expression is required, it would be

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