Respuesta :
Answer:
The dimensions of the box that minimize the materials used is [tex]6\times 3\times 2\ ft[/tex]
Step-by-step explanation:
Given : An open top box is to be built with a rectangular base whose length is twice its width and with a volume of 36 ft³.
To find : The dimensions of the box that minimize the materials used ?
Solution :
An open top box is to be built with a rectangular base whose length is twice its width.
Here, width = w
Length = 2w
Height = h
The volume of the box V=36 ft³
i.e. [tex]w\times 2w\times h=36[/tex]
[tex]h=\frac{18}{w^2}[/tex]
The equation form when top is open,
[tex]f(w)=2w^2+2wh+2(2w)h[/tex]
Substitute the value of h,
[tex]f(w)=2w^2+2w(\frac{18}{w^2})+2(2w)(\frac{18}{w^2})[/tex]
[tex]f(w)=2w^2+\frac{36}{w}+\frac{72}{w}[/tex]
[tex]f(w)=2w^2+\frac{108}{w}[/tex]
Derivate w.r.t 'w',
[tex]f'(w)=4w-\frac{108}{w^2}[/tex]
For critical point put it to zero,
[tex]4w-\frac{108}{w^2}=0[/tex]
[tex]4w=\frac{108}{w^2}[/tex]
[tex]w^3=27[/tex]
[tex]w^3=3^3[/tex]
[tex]w=3[/tex]
Derivate the function again w.r.t 'w',
[tex]f''(w)=4+\frac{216}{w^3}[/tex]
For w=3, [tex]f''(3)=4+\frac{216}{3^3}=12 >0[/tex]
So, it is minimum at w=3.
Now, the dimensions of the box is
Width = 3 ft.
Length = 2(3)= 6 ft
Height = [tex]\frac{18}{3^2}=2\ ft[/tex]
Therefore, the dimensions of the box that minimize the materials used is [tex]6\times 3\times 2\ ft[/tex]
