Answer:
The height of the highest point of the arch is 3 feet.
Step-by-step explanation:
The complete question is:
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:
![y= -0.18(x-6)(x+6)\\y=-0.18(x^{2}-36)\\y=-0.18x^{2}+6.48x](https://tex.z-dn.net/?f=y%3D%20-0.18%28x-6%29%28x%2B6%29%5C%5Cy%3D-0.18%28x%5E%7B2%7D-36%29%5C%5Cy%3D-0.18x%5E%7B2%7D%2B6.48x)
The equation is of a parabolic arch.
The general equation of a parabolic arch is:
![y=ax^{2}+bx+c](https://tex.z-dn.net/?f=y%3Dax%5E%7B2%7D%2Bbx%2Bc)
So,
a = -0.18
b = 6.48
c = 0
Highest point of the parabolic arch is the vertex of the parabolic equation if <em>a</em> < 0
.
As <em>a</em> = -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:
![x=-\frac{b}{2a}](https://tex.z-dn.net/?f=x%3D-%5Cfrac%7Bb%7D%7B2a%7D)
⇒
![x=-\frac{6.48}{2\times (-0.18)}\\\\x=18](https://tex.z-dn.net/?f=x%3D-%5Cfrac%7B6.48%7D%7B2%5Ctimes%20%28-0.18%29%7D%5C%5C%5C%5Cx%3D18)
Compute the value of <em>y</em> as follows:
![y=-0.18x+6.48](https://tex.z-dn.net/?f=y%3D-0.18x%2B6.48)
![=(-0.18\times 18)+6.48\\=3.24\\\approx 3](https://tex.z-dn.net/?f=%3D%28-0.18%5Ctimes%2018%29%2B6.48%5C%5C%3D3.24%5C%5C%5Capprox%203)
Thus, the height of the highest point of the arch is 3 feet.