Question

A rectangular tank that is 2048 ft cubed with a square base and open top is to be constructed of sheet steel of a given thickness. Find the dimensions of the tank with minimum weight.

Answer 1

Answer:

The box should have base 16ft by 16ft and height 8ft Therefore,dimensions are 16 ft by 16 ft by 8 ft

Step-by-step explanation:

We were given the volume of the tank as, 2048 cubic feet.

Form minimum weight, the surface area must be minimum.

Let the height be h and the lengths be x

the volume will be: V=x²h then substitute the value of volume, we have

2048=hx²

hence

h=2048/x²

Since the amount of material used is directly proportional to the surface area, then the material needs to be minimized by minimizing the surface area.

The surface area of the box described is

A=x²+4xh

Then substitute h into the Area equation we have

A= x² + 4x(2048/x²)

A= x² + 8192/x

We want to minimize

A

dA/dx = -8192/x² + 2 x= 0 for max or min

when dA/dx=0

dA/dx= 2x-8192/x²=0

2x=8192/x²

Hence

2x³=8192

x³=4096

x=₃√(4096)

X=16ft

Then h=2048/x²

h=2048/16²

h=8ft

The box should have base 16ft by 16ft and height 8ft

Hence the dimensions are 16 ft by 16 ft by 8 ft