Answer:
Part 1) The expression for the perimeter is [tex]P=4(25z-3)[/tex] or [tex]P=100z-12[/tex]
Part 2) The perimeter when z = 15 ft. is [tex]P=1,488\ ft[/tex]
Step-by-step explanation:
Part 1)
we have
[tex]625z^{2}-150z+9[/tex]
Find the roots of the quadratic equation
Equate the equation to zero
[tex]625z^{2}-150z+9=0[/tex]
Complete the square
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]625z^{2}-150z=-9[/tex]
Factor the leading coefficient
[tex]625(z^{2}-(150/625)z)=-9[/tex]
[tex]625(z^{2}-(6/25)z)=-9[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]625(z^{2}-(6/25)z+(36/2,500))=-9+(36/4)[/tex]
[tex]625(z^{2}-(6/25)z+(36/2,500))=0[/tex]
Rewrite as perfect squares
[tex]625(z-6/50)^{2}=0[/tex]
[tex]z=6/50=0.12[/tex] -----> root with multiplicity 2
so
The area is equal to
[tex]A=625(z-0.12)(z-0.12)=[25(z-0.12)][25(z-0.12)]=(25z-3)^{2}[/tex]
The length side of the square is [tex]b=(25z-3)[/tex]
therefore
The perimeter is equal to
[tex]P=4b[/tex]
[tex]P=4(25z-3)[/tex]
[tex]P=100z-12[/tex]
Part 2) Find the perimeter when z = 15 ft.
we have
[tex]P=100z-12[/tex]
substitute the value of z
[tex]P=100(15)-12=1,488\ ft[/tex]
