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=[tex](19-2x)[/tex] inches
Width of open box=[tex]4-2x inches[/tex]
Height of box=[tex]x inches[/tex]
Volume of box=[tex]l\times b\times h[/tex]
Substitute the values then we get
Volume of box=[tex]x(19-2x)(4-2x)[/tex]
[tex]x(19-2x)(4-2x) > 0[/tex]
Because volume of open box is always greater than zero.
Then , [tex]x >0[/tex]
Substitute
[tex]19-2x > 0[/tex]
[tex]2x-19 < 0[/tex]
[tex]2x < 19[/tex] By adding 19 on both sides
Dividing by 2 on both sides
[tex]x< \frac{19}{2}=9.5[/tex]
It means [tex]x\in (0,9.5)[/tex]
Substitute [tex]4-2x> 0[/tex]
[tex]2x-4< 0[/tex]
Adding 4 on both sides then we get
[tex]2x < 4[/tex]
Dividing by 2 on both sides
[tex]x< \frac{4}{2}=2[/tex]
[tex]x <2[/tex]
Then, [tex]x\in (0,2)[/tex]
Therefore, domain of the box =[tex](0,2)\cap (0,9.5)=(0,2)[/tex]
Then ,a=0 and b=2
