Question

A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 1458 cubic feet. The concrete for the base costs ​$4 per square​ foot, the material for the roof costs ​$6 per square​ foot, and the material for the sides costs ​$2.50 per square foot. Find the dimensions of the most economical shed.

Answer 1

the length of base and height of shed are 9 cm and 18 cm respectively.

Explanation:

let the length of the base be = x ft ( the base is square)

and height be = y ft

given volume o the shed = 1458 cm^3

⇒
`x^2y= 1458`

also, according to question total cost C

`C= 4x^2+6x^2+4xy*2.5`

`C= 10x^(2)+ 10x* y`

now put y= 1428/x^2 in the above equation

`C= 10 x^(2)+(14580)/(x)`

now differentiating we get

`C'=  20 x-(14580)/(x^(2))`

now set it equal to zero for most economical cost

`20 x =(14580)/(x^(2))`

on calculating we get x= 9 cm

and y= 18 cm

therefore, the length of base and height of shed are 9 cm and 18 cm respectively.