we know that
The hallway is painted using equal amounts of white and black paint
so
To know the amount of white paint------> calculate the area of white paint
[tex]A=5x*6=30x\ ft^{2}[/tex]
To know the amount of black paint------> calculate the area of black paint
[tex]A=(x+1)*4*6=(24+24x)\ ft^{2}[/tex]
equate the areas
[tex]30x=(24+24x)\\30x-24x=24\\6x=24\\x=4\ ft[/tex]
Find the length of the hallway
[tex]5x+(4x+4)=9x+4=9*4+4=40\ ft[/tex]
therefore
the answer part a) is
The hallway is [tex]40\ ft[/tex] long
Part b) Can this same hallway be painted with the same pattern, but using twice as much black paint as white paint?
we know that
the amount of white paint is [tex]30x\ ft^{2}[/tex]
if the black paint is twice that the white paint
then
amount of black paint is [tex]2*30x=60x\ ft^{2}[/tex]
amount of black paint with the same pattern is [tex](24+24x)\ ft^{2}[/tex]
equate
(24+24x)\ ft^{2}=(60x)\ ft^{2}
solve for the new value of x
[tex]24x+24=60x\\ 60x-24x=24\\ x=24/36\\x= 2/3\ ft[/tex]
Find the new length of the hallway
[tex]5x+(4x+4)=9x+4=9*(2/3)+4=10\ ft[/tex]
therefore
the answer is
Yes, the same hallway can be painted with the same pattern, using twice as much black paint as white paint, but the hallway decreases in length
