A dome tent’s arch is modeled by y= -0.18(x-6)(x+6)where x and y are measured in feet.

A dome tent’s arch is modeled by y= -0.18(x-6)(x+6) where x and y are measured in feet. To the nearest foot, what is the height of the highest point of the arch.

Solution:

The expression provided is:

[tex]y= -0.18(x-6)(x+6)\\y=-0.18(x^{2}-36)\\y=-0.18x^{2}+6.48x[/tex]

The equation is of a parabolic arch.

The general equation of a parabolic arch is:

[tex]y=ax^{2}+bx+c[/tex]

So,

a = -0.18

b = 6.48

c = 0

Highest point of the parabolic arch is the vertex of the parabolic equation if a < 0 .

As a = -0.18 < 0, the ordinate of vertex of equation will give the height of highest point of arch.

For a parabola the abscissa of vertex is given as follows:

[tex]x=-\frac{b}{2a}[/tex]

⇒

[tex]x=-\frac{6.48}{2\times (-0.18)}\\\\x=18[/tex]

Compute the value of y as follows:

[tex]y=-0.18x+6.48[/tex]

[tex]=(-0.18\times 18)+6.48\\=3.24\\\approx 3[/tex]

Thus, the height of the highest point of the arch is 3 feet.