The dimension of the container of this size that has the minimum cost is
,
and ![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
The given parameters are:
- Volume = 480m^3
- Cost: $5 per square meter for the bottom, and $3 per square meter for the sides.
Assume the dimensions of the container are: x, y and z.
The volume (V) would be:

Substitute 480 for V

And the objective cost function would be:

Differentiate the cost function using Lagrange multipliers



Divide equation (1) by (2)


Factor out 2

Cross multiply

Evaluate the like terms

Divide both sides by 3z

Divide the first equation by the third


Factor out 2

Cross multiply

Cancel out the common terms

Divide both sides by y

Make z the subject

So, we have:
and 
Recall that:

Substitute
and 

So, we have:

Multiply both sides by 3/4

Take the cube roots of both sides
![x = \sqrt[3]{360}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%5B3%5D%7B360%7D)
Recall that:

So, we have:
![y = \sqrt[3]{360}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%5B3%5D%7B360%7D)
Also, we have:

So, we have:
![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
Hence, the dimension of the container of this size that has the minimum cost is
,
and ![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
Read more about Lagrange multipliers at:
brainly.com/question/4609414