Minimizing surface area. Mendoza Soup Company is constructing an open-top, squar
ID: 2877595 • Letter: M
Question
Minimizing surface area. Mendoza Soup Company is constructing an open-top, square-based, rectangular metal lank that will have a volume of 32 ft^3. What dimensions will minimize surface area? What is the minimum surface area? Minimizing surface area. Drum Tight Containers is designing an open-top, square-based, rectangular box that will have a volume of 62.5 in^3 What dimensions will minimize surface area? What is the minimum surface area? Minimizing surface area. Open Air Waste Management is designing a rectangular construction dumpster that will he twice as long as it is wide and must hold 122 yd^3 of debris. Find the dimensions of the dumpster that will minimize us surface area.Explanation / Answer
Let the measurements of the box be x, y ,z
Volume = xyz= 32 (given ) S urface area = xy + 2yz +2xz and z = 32 / xy
SA = f =xy + 2y(32/xy) +2x( 32/yx) = xy + 64 /x + 64 /y to find the min SA find the partial derivatives
f x = y - 64 / x2 , fy = x - 64 / y2 equating these terms to 0 we get the critical points
-0 => x2 y= 64 , y2 x =64 => x=y and multiplying the 2 terms we get x3 y3 =64x64 =>
x=4 ,y=4 -----(1)
fxx= - 128 / x3 , fyy= -128 / y3 , fxy= 1
Evaluating fxx fyy - fxy2 = >0 at (4,4) hence the values x=4 , y=4 will give min value for Surface Area
xyz=32 => z=2 when x=4 ,y=4 snd the SA = 16+ 16+16 = 48 sq ft