```Question 200168
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/a}}}
:
Substitute (180/a) in the pythag equation
a^2 + ({{{180/a}}})^2 = 400
:
a^2 + {{{32400/(a^2)}}} = 400
Multiply equation by a^2:
a^4 + 32400 = 400a^2
a^4 - 400a^2 + 32400 = 0
:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
in this equation a=1, b=-400, c=32400; solve for x^2
{{{x^2 = (-(-400) +- sqrt(-400^2 - 4 * 1 * 32400 ))/(2*1) }}}
:
{{{x^2 = (400 +- sqrt(160000 - 129600 ))/2 }}}
:
{{{x^2 = (400 +- sqrt(30400 ))/2 }}}
Two solutions
{{{x^2 = (400 + 174.346 )/2 }}}
{{{x^2 = 574.356/2}}}
x^2 = 287.178
x = {{{sqrt(287.178)}}}
x = 16.946" the length of the rectangle (a)
and
{{{x^2 = (400 - 174.346 )/2 }}}
:
{{{x^2 = 225.654/2}}}
x^2 = 112.827
x = {{{sqrt(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 = {{{sqrt(16.946^2 + 10.622^2)}}}
d = 20.02 ~ 20"
```