5. A rectangular-box-shaped cargo container, open at the top, is to be construct

Question

5. A rectangular-box-shaped cargo container, open at the top, is to be constructed from sheet steel so as to have a finished volume of 32m2. To be useful to shippers, no edge dimension of the container can be smaller than lm Set up a suitable 2D max/min problem to find the dimensions of the minimum cost container Your setup must include an explicit cost function and an explicit domain over which the cost function will be optimized. Solve the problem in part (a) to determine the optimal dimensions. Your solution must include analysis of the boundary of the domain.

Explanation / Answer

Open at the top...

So, total area= xy + 2yz+ 2xz

f : A = xy + 2yz + 2xz ---> to be minimized
g : V = xyz = 32

So,
finding partials :

fx = y + 2z
fy = x + 2z
fz = 2x+ 2y

gx = yz
gy = xz
gz = xy

Linkin em up :
fx = mgx , fy = mgy and fz = mgz....

y + 2z = m(yz)
x + 2z = m(xz)
2x + 2y = m(xy)

Divide first two :
(y + 2z)/(x + 2z) = y/x
Crossmultiply :
xy + 2xz = xy + 2yz
Cancel xy :
2xz = 2yz
Thus, x = y

Divide second and third equations :
(x+2z)/(2x+2y) = z/y
Crossmultiply :
xy + 2yz = 2xz + 2yz
Cancel 2yz :
xy = 2xz
y = 2z

So, we have
x = y
y = y
z = y/2

Using volume, we have
xyz = 32

Sub in terms of y :
y * y * y/2 = 32

y^3 / 2 = 32

y^3 = 64

y = 4

And thus, x= 4
y = 4
z = 2

So, length = 4
Width = 4
Height = 2

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