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=$l\times b\times h$

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

$x< \frac{19}{2}=9.5$

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

$x< \frac{4}{2}=2$

$x$

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

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

Then ,a=0 and b=2