SOLUTION: A rounded metal plate is made from a rectangular sheet of metal of size x cms by y cms. Each corner is rounded off to a quadrant of circle, radius r.
a) If A (square cms) is the
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a) If A (square cms) is the
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Question 1191686: A rounded metal plate is made from a rectangular sheet of metal of size x cms by y cms. Each corner is rounded off to a quadrant of circle, radius r.
a) If A (square cms) is the area of the finished plate,
find an expression for A in terms of x,y and r, explaining your method.
b) Hence show that r=sqrt[ (xy-A)/(4-pi) ] Found 2 solutions by ankor@dixie-net.com, Edwin McCravy:Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website! A rounded metal plate is made from a rectangular sheet of metal of size x cms by y cms.
Each corner is rounded off to a quadrant of circle, radius r.
a) If A (square cms) is the area of the finished plate, find an expression for A in terms of x,y and r, explaining your method.
The area removed from each corner:
The area of the metal plate = xy
:
A =
b) Hence show that r=sqrt( xy-A/4-pi) = xy - A
distribute the 4 = xy - A
factor out r^2 = xy - A
divide by (4-pi)
r^2 =
:
r =
We start with a rectangle which has area
Then we round off the corners, which means that we
subtract the 4 little corners that we round off from
the original area which is .
Next we draw in the quadrants of a circle of radius r to help us
see how much was subtracted from the corners in the rounding
process.
Now we look at just those quadrants of a circle:
Now we put the four together and have a circle inscribed in a square
Now we can see how much we have to subtract off from the
area of the original rectangle whose area is
We have the area of a square whose side is twice the radius r.
That's . From that we subtract the area of the
circle , and get .
That means the area of the rounded metal plate is xy minus
the area of those corners.
We can factor out the r2 in the last two terms:
<--expression for A
---------------------------------
To get your answer for r, go back to the previous step:
Edwin
