**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**

**Derivate w.r.t x,
**

**The critical point when V'(x)=0
**

**Solve by quadratic formula,**

**Derivate again w.r.t x,
**

**Now,** (+ve)

(-ve)

**So,** there is maximum at x=1.392.

**The length of the box is **

**The breadth of the box is **

**The height of the box is** h=1.392.

**The dimension of the open rectangular box is ****.**

The volume of the box is

**The volume of the box is 8.217 cubic inches.**