A rectangular tank with a square base, an open top. and a volume of 27.436 ft^3

Question

A rectangular tank with a square base, an open top. and a volume of 27.436 ft^3 is to be constructed of sheet steel Find the dimensions of the tank that has the minimum suffice area. Let s be the length of one of the sides of the square base and let A be the surface area of the tank Write the objective function. The interval of interest of the objective (unction is (Simplify your answer Type your answer in interval notation) The tank with the minimum surface area has a height of ft and a square base with a side length Of ft

Explanation / Answer

Solution:

The length and width are x and the height is y,

so we have:
x2 y = 27436
y = 27436/(x2)

The area of the base is x2

The area of each side is xy, and there are 4 sides,

So, the total sides of area is 4xy

Surface area:
A = x2 + 4xy

A = x2 + 4x(27436/(x2))

A = x2 + (109744 / x)

dA/dx=2x-109744/x2

dA/dx=(2x3-109744)/x2

dA/dx=0

when

2x3=109744

x3=54872

x=38 ft

h=27436/x2= 19ft. (x2 = 1444)

A(38)=x2+109744/x=4332 ft^3 (this is the minimum area)

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