A cardboard box without a lid is to be made with a volume of 44 ft3. Find the dimensions of the box that requires the least amount of cardboard.
Accepted Solution
A:
Answer:x = 3.53 fty - 3.53 ftz = 3.53 ftStep-by-step explanation:given details volume = 44 ft^3let cardboard dimension is x and y and height be zwe know that area of given cardboard without lid is given as A = xy + 2xy + 2yzxyz = 44 ft^3To minimize area we haveA = xy + 2x (44/xy) + 2y(44/xy) A = xy + (44/y) + (44/x)we have[tex]Ax = y - \frac{44}{x^2}[/tex][tex]0 = yx^2 = 44[/tex]................1[tex]Ay = x - \frac{44}{y^2}[/tex][tex]0 = x - \frac{44}{y^2}[/tex][tex]xy^2 = 44[/tex] ..............2from 1 and 2[tex]yx^2 = xy^2[/tex]xy(y-x) = 0 xy = 0 or y = xfrom geometry of probelem x ≠ 0 and y ≠ 0so y = xx^3 = 44x = 3.53 ft = yz = 44/xy = 3.53