Answer:
The dimension of the open rectangular box is
.
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 ![V=l\times b\times h](https://tex.z-dn.net/?f=V%3Dl%5Ctimes%20b%5Ctimes%20h)
![V=(11-2x)\times (7-2x)\times x](https://tex.z-dn.net/?f=V%3D%2811-2x%29%5Ctimes%20%287-2x%29%5Ctimes%20x)
![V=4x^3-36x^2+77x](https://tex.z-dn.net/?f=V%3D4x%5E3-36x%5E2%2B77x)
Derivate w.r.t x,
![V'(x)=4(3x^2)-2(36x)+77](https://tex.z-dn.net/?f=V%27%28x%29%3D4%283x%5E2%29-2%2836x%29%2B77)
![V'(x)=12x^2-72x+77](https://tex.z-dn.net/?f=V%27%28x%29%3D12x%5E2-72x%2B77)
The critical point when V'(x)=0
![12x^2-72x+77=0](https://tex.z-dn.net/?f=12x%5E2-72x%2B77%3D0)
Solve by quadratic formula,
![x=\frac{18+\sqrt{93}}{6},\frac{18-\sqrt{93}}{6}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B18%2B%5Csqrt%7B93%7D%7D%7B6%7D%2C%5Cfrac%7B18-%5Csqrt%7B93%7D%7D%7B6%7D)
![x=4.607,1.392](https://tex.z-dn.net/?f=x%3D4.607%2C1.392)
Derivate again w.r.t x,
![V''(x)=24x-72](https://tex.z-dn.net/?f=V%27%27%28x%29%3D24x-72)
Now,
(+ve)
(-ve)
So, there is maximum at x=1.392.
The length of the box is ![l=11-2x](https://tex.z-dn.net/?f=l%3D11-2x)
![l=11-2(1.392)=8.216](https://tex.z-dn.net/?f=l%3D11-2%281.392%29%3D8.216)
The breadth of the box is ![b=7-2x](https://tex.z-dn.net/?f=b%3D7-2x)
![b=7-2(1.392)=4.216](https://tex.z-dn.net/?f=b%3D7-2%281.392%29%3D4.216)
The height of the box is h=1.392.
The dimension of the open rectangular box is
.
The volume of the box is ![V=l\times b\times h](https://tex.z-dn.net/?f=V%3Dl%5Ctimes%20b%5Ctimes%20h)
![V=8.216\times 4.216\times 1.392](https://tex.z-dn.net/?f=V%3D8.216%5Ctimes%204.216%5Ctimes%201.392)
![V=48.217\ in.^3](https://tex.z-dn.net/?f=V%3D48.217%5C%20in.%5E3)
The volume of the box is 8.217 cubic inches.