Respuesta :
Answer:
Therefore the dimension of box is 7 ft by 7 ft by 14 ft.
Step-by-step explanation:
Given that, a storage shade is be built in the shape of a box with a square base.
Let the height of the box be h and the length of one side of the square base be x.
The area of the square base is = side²
=x²
The volume of the box is = area of the base × height
=x²h
According to the problem,
x²h=686
[tex]\Rightarrow h=\frac{686}{x^2}[/tex] .......(1)
The concrete for the base costs $5 per square foot.
The material for the base costs =$ 5x²
The material for the roof costs $9 per square foot.
The material cost for roof is =$9x²
The material for the sides costs $3.50 per square foot.
The material cost for sides =$(3.50× 4xh )
=$14xh
Total cost =$(5x²+9x²+14xh)
=$(14x²+14 xh)
Let
C = 14x²+14 xh
Putting [tex]h=\frac{686}{x^2}[/tex]
[tex]C=14x^2+14 x.\frac{686}{x^2}[/tex]
[tex]\Rightarrow C=14x^2+\frac{9604}{x}[/tex]
Differentiating with respect to x
[tex]C'= 28x-\frac{9604}{x^2}[/tex]
Again differentiating with respect to x
[tex]C''= 28+\frac{19208}{x^2}[/tex]
To find the dimension set C'=0
[tex]28x-\frac{9604}{x^2}=0[/tex]
[tex]\Rightarrow 28x=\frac{9604}{x^2}[/tex]
[tex]\Rightarrow x^3=\frac{9604}{28}[/tex]
[tex]\Rightarrow x=7[/tex]
Now,
[tex]C''|_{x=7}= 28+\frac{19208}{7^2}>0[/tex]
Since at x=7, C''>0, So at x=7 , The cost of material will be minimum.
The height of the box is [tex]h=\frac{686}{x^2}[/tex]
[tex]=\frac{686}{7^2}[/tex]
=14 foot
Therefore the dimension of box is 7 ft by 7 ft by 14 ft.
The cost of the material is [tex]=14x^2+\frac{9604}{x}[/tex]
[tex]=14(7)^2+\frac{9604}{7}[/tex]
=$2,058
