Respuesta :
Answer:
The dimension of the open rectangular box is [tex]8.216\times 4.216\times 1.392[/tex].
The volume of the box is 8.217 cubic inches.
Step-by-step explanation:
Given : The open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by cutting congruent squares from the corners and folding up the sides.
To find : The dimensions and the volume of the open rectangular box ?
Solution :
Let the height be 'x'.
The length of the box is '11-2x'.
The breadth of the box is '7-2x'.
The volume of the box is [tex]V=l\times b\times h[/tex]
[tex]V=(11-2x)\times (7-2x)\times x[/tex]
[tex]V=4x^3-36x^2+77x[/tex]
Derivate w.r.t x,
[tex]V'(x)=4(3x^2)-2(36x)+77[/tex]
[tex]V'(x)=12x^2-72x+77[/tex]
The critical point when V'(x)=0
[tex]12x^2-72x+77=0[/tex]
Solve by quadratic formula,
[tex]x=\frac{18+\sqrt{93}}{6},\frac{18-\sqrt{93}}{6}[/tex]
[tex]x=4.607,1.392[/tex]
Derivate again w.r.t x,
[tex]V''(x)=24x-72[/tex]
Now, [tex]V''(4.607)=24(4.607)-72=38.568>0[/tex] (+ve)
[tex]V''(1.392)=24(1.392)-72=-38.592<0[/tex] (-ve)
So, there is maximum at x=1.392.
The length of the box is [tex]l=11-2x[/tex]
[tex]l=11-2(1.392)=8.216[/tex]
The breadth of the box is [tex]b=7-2x[/tex]
[tex]b=7-2(1.392)=4.216[/tex]
The height of the box is h=1.392.
The dimension of the open rectangular box is [tex]8.216\times 4.216\times 1.392[/tex].
The volume of the box is [tex]V=l\times b\times h[/tex]
[tex]V=8.216\times 4.216\times 1.392[/tex]
[tex]V=48.217\ in.^3[/tex]
The volume of the box is 8.217 cubic inches.
