Answer:
Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.
Therefore the minimum cost to manufacture the box is $12936.
Step-by-step explanation:
Given the volume of the open box is 196 cube ft.
Let the one side of the base of the box be x.
Since the base of the box is square in shape.
The other side of the base is also x.
The area of the base of the box = x² square ft
Let the height of the box be h.
The volume of the box is = Area of the base× height
=(x²×h) cube ft.
=x²h cube ft.
According to the problem,
x²h=196
The material cost for the bottom is $88 per square ft and the material cost for the sides $77 per square ft.
The material cost for the bottom of the box = $(88×x²)
=$88x²
The area sides = 2(length+width)height
=2(x+x)h
=4xh square ft.
The material cost for the sides of the box =$(77×4xh)
=$308 xh
Total cost to make the box is =$(88x²+308 xh)
∴C=88x²+308 xh [ C is in dollar]
Now putting
Differentiating with respect to x
Again differentiating with respect to x
Now to find the minimum cost, we set C'=0
Now,
Therefore at x=7, the manufacturing cost will be minimum.
The height of the box is = 4ft
Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.
The manufacturing cost is [ putting x=7]
=$12936
Therefore the minimum cost to manufacture the box is $12936.