Question

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 1

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 Arrange as a quadratic equation a^4 - 400a^2 + 32400 = 0 : Use the quadratic formula {{{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"