Answer:
length=10 ft , width = 10 ft and height = 5 ft
Step-by-step explanation:
using Lagrange multipliers , we have the main function that is the Area A of the tank ,
A(x,y,z) = x*y + 2*x*z + 2*y*z
constrained to the Volume V(x,y,z) = x*y*z=a = 500 ft³
using Lagrange multipliers
Ax - λ*Vx = 0 → (y + 2*z) - y*z*λ = 0 → λ= 1/z + 2/y
Ay - λ*Vy = 0 → (x + 2*z) - x*z*λ = 0 → λ= 1/z + 2/x
Az - λ*Vz = 0 → (2*x + 2*y) - x*y*λ = 0 → λ= 2/x + 2/y
V =a → x*y*z=a
adding the first and second equations
2*λ= 1/z + 2/y + 1/z + 2/x = 2/z + λ
λ = 2/z → z= 2/λ
therefore
λ= 1/z + 2/y = λ/2 + 2/y
λ/2 = 2/y → y= 4/λ
and similarly x=4/λ
then
x*y*z=a
2/λ*4/λ*4/λ= a
32/λ³ = a
λ = ∛32/a
therefore
z= 2/λ = 2*∛a/32 = 2*∛(500/32) = 5
y= 4/λ = 2*5 = 10
x=10
therefore the dimensions that minimize the area are x=10 , y=10 and z=5 (length=10 ft , width = 10 ft and height = 5 ft)
