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
.......(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
Differentiating with respect to x
Again differentiating with respect to x
To find the dimension set C'=0
Now,
Since at x=7, C''>0, So at x=7 , The cost of material will be minimum.
The height of the box is
=14 foot
Therefore the dimension of box is 7 ft by 7 ft by 14 ft.
The cost of the material is
=$2,058