Question

A shipping company must design a closed rectangular shipping crate with a square base. The volume is 2880ft3. The material for the top and sides costs $3 per square foot and the material for the bottom costs $7 per square foot. Find the dimensions of the crate that will minimize the total cost of material.

Answer 1

The dimensions of the crate that will minimize the total cost of material are 10, 10, and 29.

How to compute the dimension?

The volume will be:

= Length × Width × Height

= lwh

For the square base, w = l

2880 = w²h

The cost function will be:

C = 12wl + 4wh + 4lh

Put l = w.

C = 12w² + 8wh

C = 12w² + 8w(2880/w²)

C = 12w² + 23040/w

Differentiate with respect to w

C'(w) = 0

24w = 23040/w²

24w³ = 23040

w³ = 960

w = 10 approximately

The length is 16 as well as they are the same. The height will be:

= 2880/w²

= 2880/10²

= 28.8 = 29

Therefore, the dimensions are 10, 10, and 29.

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