Question

A square-based shipping crate is being designed that must contain a volume of 16 ft3 . The material that is used for the base and the lid costs 3 dollars/ft2 , while the material used for the sides costs 2 dollars/ft2 . What are the most cost-effective dimensions of such a crate?

Answer 1

Step-by-step explanation:

Given

volume
`V=16 ft^3`

Suppose base is square with side L

height of crate is h

Volume
`V=L^2* h`

`16=L^2* h`

Cost of top and bottom area
`c_1=3L^2`

Cost of Side area
`c_2=4Lh* 2=8Lh=8L* (16)/(L^2)=(128)/(L)`

Total Cost
`C=c_1+c_2`

Total Cost
`C=3L^2+(128)/(L)`

Differentiate C w.r.t Length

`(dC)/(dL)=6L-(128)/(L^2)`

`L^3=(128)/(6)`

`L=2.75 ft`

`h=(16)/(2.75^2)=11.46 ft`

Dimensions are
`L* L* h=2.75* 2.75* 11.46`