Question

An open box is to be made from a flat piece of material 19 inches long and 4 inches wide by cutting equal squares of length x from the corners and folding up the sides. Write the volume Vof the box as a function of x. Leave it as a product of factors, do not multiply out the factors. V= If we write the domain of the box as an open interval in the form (a,b), then what is a=? a= and what is b=? b=

Answer 1

Answer with Step-by-step explanation:

We are given that

Length of flat piece of material =19 inches

Width of flat piece=4 inches

An open box is to be made from a flat piece by cutting equal squares of length x from the corners and folding up the sides

Therefore, length of open box=
`(19-2x)` inches

Width of open box=
`4-2x inches`

Height of box=
`x inches`

Volume of box=
`l* b* h`

Substitute the values then we get

Volume of box=
`x(19-2x)(4-2x)`

`x(19-2x)(4-2x) > 0`

Because volume of open box is always greater than zero.

Then ,
`x >0`

Substitute

`19-2x > 0`

`2x-19 < 0`

`2x < 19` By adding 19 on both sides

Dividing by 2 on both sides

`x< (19)/(2)=9.5`

It means
`x\in (0,9.5)`

Substitute
`4-2x> 0`

`2x-4< 0`

Adding 4 on both sides then we get

`2x < 4`

Dividing by 2 on both sides

`x< (4)/(2)=2`

`x <2`

Then,
`x\in (0,2)`

Therefore, domain of the box =
`(0,2)\cap (0,9.5)=(0,2)`

Then ,a=0 and b=2