Answer:
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
