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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 567567 cubic feet. The concrete for the base costs ​$22 per square​ foot, the material for the roof costs ​$55 per square​ foot, and the material for the sides costs ​$4.504.50 per square foot. Find the dimensions of the most economical shed.

Respuesta :

Answer:

length of the base = 9 ft

and height of the shed = 7 ft

Step-by-step explanation:

given data

volume =  567 cubic feet

base costs = ​$2 per square​ foot

roof costs ​= $5 per square​ foot

sides costs ​$4.50 per square foot

solution

we take here length of the base = x ft

and height of the shed = y ft

so Volume will be express as

volume = x²× y

567 =  x² × y

y =  [tex]\frac{567}{x^2}[/tex]

and

we know  cost of material is express as here

cost of material  = cost of base + cost top + cost 4 side   ..................1

put here value

cost  = x²(2) + x²(5)  + 4xy (4.5)

cost = 7x² + 18xy

put here y  value

cost =   7x² + 18 x ( [tex]\frac{567}{x^2}[/tex] )

differentiate and  we get

C' = 14x - [tex]\frac{10206}{x^2}[/tex]  

we put here C' = 0 and we get

14x - [tex]\frac{10206}{x^2}[/tex]   = 0

solve it we get

x = 9 ft

and

y = [tex]\frac{567}{9^2}[/tex]  

y = 7 ft