A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.

Length = 10

Width = 10

Height = 5

Surface Area_minimum = 300 ft²

Step-by-step explanation:

Given the data in the question;

Let x represent  the length, y represent the width and z represent the height.

Volume of tank = length × width × height

V = xyz

so

Volume V = 500 = xyz

Tank with no Top;

Surface area = length × width + 2 × height( length + width )

which is;

S = xy + 2z( x + y )

f( x,y ) = xy + 1000/y + 1000/x

f"x = y - 1000/x²;     fy = x - 100/y²

Critical Value

x²y = 1000;    xy² = 1000

Critical Pont:  ( 10, 10 )

Fx"x = 2000/x³;    Fyy = 2000/y³;   Fxy = 1

D( x,y ) = Fx"xFyy - ( Fxy )²

D(10, 10): 4-1 = 3 > 0  Fx"x > 0

so surface area minimum ( 10, 10 )

Length = 10

Width = 10

Height = 5

Surface Area_minimum S = xy + 2z( x + y )

S = 10×10 + 2×5( 10 + 10 )

S = 100 + 10(20)

S = 100 + 200

S = 300 ft²

Therefore;

Length = 10

Width = 10

Height = 5

Surface Area_minimum = 300 ft²