A rectangular tank with a square base, an open top, and a volume of 6912ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

The dimensions of the rectangular tank is 30.24 ft by  30.24 ft by 7.56 ft.

Step-by-step explanation:

Given that,

A rectangular tank  is to be constructed with a square base and a volume of 6912 ft³.

Let length of the one side of the base be x and the height of the tank be y.

Then, the volume of the tank is= area of the base × height

                                                    =(x²)×y

                                                    =x²y ft²

Then,

x²y = 6912

The surface area of the tank is

= surface area of the sides + surface area of the base

=4(xy)+x²           [ surface area of each wall= length × width =xy]

=4xy+x²

A= 4xy + x²

Plug in the above expression

Differentiating with respect to x

Again differentiating with respect to x

For maximum or minimum, A'=0

⇒x ≈ 30.24

Now

So, at x=30.24 ft the surface area minimum.

The length of one side of the base of the tank is = 30.24 ft

Then the height of the rectangular tank is

 =7.56 ft

The dimensions of the rectangular tank is 30.24 ft by  30.24 ft by 7.56 ft.

cheater

step-by-step explanation: