A rectangular sheet metal w/ an area of 1200 in虏 is bent into a cylinder w/ volume of 600 in鲁. Dimensions?

We are looking for the dimensions of the original rectangular sheet metal, not the new cylindrical ';stovepipe';. Please show work so that I will know how to solve this problem.





I AlWAYS select the best answer. Thank you.A rectangular sheet metal w/ an area of 1200 in虏 is bent into a cylinder w/ volume of 600 in鲁. Dimensions?
The dimensions of the rectangular sheet are x inches by y inches





xy = 1200


y = 1200/x





The cylinder will have a height of x inches and the circumference of the base is y inches





y = 2蟺 r





Volume of cylinder is [area of the base]*(height)





V = 蟺 r^2 x = 1200 in^3





Let's find an expression for the radius in terms of x





the circumference = y = 1200/x = 2蟺 r





so





r = 1200/(2蟺 x) = 600/(蟺 x)





The area of the base of the cylinder is 蟺 r^2





in terms of x, that will be





蟺(600)^2 / (蟺 x)^2 = (600)^2 / 蟺 x^2





Volume = x[600)^2/蟺c^2 = (600)^2 / 蟺 x = 600





Solve for x





x = 600/蟺 inches





y = 1200/x = 2蟺 inches





These are the exact dimensions of the rectangular sheet





QEDA rectangular sheet metal w/ an area of 1200 in虏 is bent into a cylinder w/ volume of 600 in鲁. Dimensions?
method:





write formulas for area and of a rectangle and volume of a cylinder, then use the problem description (bent into a cylinder) to relate cylinder length to rectangle length (equal) and cylinder radius to rectangle width (width = pi*d= 2*pi*r) This give two equations w twp unknowns. Simplify and resolve. As follows:





Let x= width y= length





area = 1200 = x*y (1)





Volume(cyl) = L*pi*r^^2= 600 (2)





L=y (';bent onto a cylinder';)





since x=circumference of cycl and circumference = pi*D(iameter) and radius = D/2





then x=pi*D=2*pi*r so r=x/(2*pi)





substitute this result for x into equation 2





600 = y*x^^2/4*pi (3)





from (1): y =1200/x





substituting this for y in (3): 600 =1200x^^2/(x*4*pi) = 300*x/pi





so x = 600*pi/300= 2*pi





from (1)





y=1200/x = 1200/2*pi= 600/pi








Rectangular sheet is width = 2*pi inches length = 600/pi inches
1) vol of cylinder = pi r^2 * h = 600


2) area of metal in cylinder = 2pir*h = 1200


dividing 1 by 2 gives:





r/2 = 600/1200 =1/2





so r = 1


length of sheet = 2 pi r = 2pi inches


other side of rectangle = 1200/2pi inches