SOLUTION: Hi! 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 re

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Question 200168: Hi!
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?
thank you for your help!
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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
b = 180%2Fa
:
Substitute (180/a) in the pythag equation
a^2 + (180%2Fa)^2 = 400
:
a^2 + 32400%2F%28a%5E2%29 = 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
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
in this equation a=1, b=-400, c=32400; solve for x^2
x%5E2+=+%28-%28-400%29+%2B-+sqrt%28-400%5E2+-+4+%2A+1+%2A+32400+%29%29%2F%282%2A1%29+
:
x%5E2+=+%28400+%2B-+sqrt%28160000+-+129600+%29%29%2F2+
:
x%5E2+=+%28400+%2B-+sqrt%2830400+%29%29%2F2+
Two solutions
x%5E2+=+%28400+%2B+174.346+%29%2F2+
x%5E2+=+574.356%2F2
x^2 = 287.178
x = sqrt%28287.178%29
x = 16.946" the length of the rectangle (a)
and
x%5E2+=+%28400+-+174.346+%29%2F2+
:
x%5E2+=+225.654%2F2
x^2 = 112.827
x = sqrt%28112.827%29
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 = sqrt%2816.946%5E2+%2B+10.622%5E2%29
d = 20.02 ~ 20"