Question

You have a open box that is made from a 16 in. x30 in. piece of cardboard. When you cut out the squares of equal size from the 4 corners and bending it. What size should the squares be to get this box to work with the largest volume?

Answer 1

`3 1/3` inches to be cut from `4` corners and bend to get
box for maximum volume of `725.93` cubic inches.

Explanation:

Card board size is `L=30 and W=16` inches

Let `x` in square is cut from `4` corners and bended into

a box whos size is now `L=30-2x , W=16-2x and h=x`

inches. Volume of the box is `V=(30-2x)(16-2x)x` cubic

inches. `V=(4x^2-92x+480)x = 4x^3-92x^2+480x`.

For maximum value `(dV)/dx=0`

`(dV)/dx=12x^2-184x+480=12(x^2-46/3x+40)`

`12(x^2-12x-10/3x+40)= 12(x(x-12)-10/3(x-12))`

or `12(x-12)(x-10/3)=0 :.` Critical points are

`x=12 ,x=10/3; x !=12` , as `24` inches cannot be removed from

`16` inches width. So `x= 10/3 or 3 1/3` inches to be cut.

Slope test may be examined at`(x=3 and x=4)` to show

volume is maximum. `(dV)/dx=12(x-12)(x-10/3)`

`(dV)/dx(3)= (+) and (dV)/dx(4)= (-)`. Slope at critical point

is from positive to negative , so the volume is maximum.

The maximum volume is `V=(30-20/3)(16-20/3)10/3`or

`V=(30-20/3)(16-20/3)10/3 ~~725.93` cubic inches. [Ans]