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A circular piece of sheet metal has a diameter of 20 in.
The edges are to be cut off to form a rectangle of area 180 in^2 (see the figure).
What are the dimensions of the rectangle
Let a = the length
Let b = the width
The diameter of 20" = the diagonal of the rectangle
a^2 + b^2 = 20^2; pythag
a^2 + b^2 = 400
Area of the rectangle
a * b = 180; the area of the rectangle
Substitute (180/a) in the pythag equation
a^2 + (
)^2 = 400
Multiply equation by a^2:
a^4 + 32400 = 400a^2
Arrange as a quadratic equation
a^4 - 400a^2 + 32400 = 0
Use the quadratic formula
in this equation a=1, b=-400, c=32400; solve for x^2
x^2 = 287.178
x = 16.946" the length of the rectangle (a)
x^2 = 112.827
x = 10.622" the width of the rectangle (b)
The rectangle is: 16.946 by 10.622
Check this by finding the area with these values
16.946 * 10.622 = 180.00 sq/in
Find the diameter/diagonal on a calc using these values; enter:
d = 20.02 ~ 20"