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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.

Respuesta :

Manetho

Answer:

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

Step-by-step 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

⇒ [tex]x^2y= 1458[/tex]

also, according to question total cost C

[tex]C= 4x^2+6x^2+4xy\times2.5[/tex]

[tex]C= 10x^{2}+ 10x\times y[/tex]

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

[tex]C= 10 x^{2}+\frac{14580}{x}[/tex]

now differentiating we get

[tex]C'=  20 x-\frac{14580}{x^{2}}[/tex]

now set it equal to zero for most economical cost

[tex]20 x =\frac{14580}{x^{2}}[/tex]

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.

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