Answer:
The parabolic shape of the door is represented by
. (See attachment included below). Head must 15.652 inches away from the edge of the door.
Step-by-step explanation:
A parabola is represented by the following mathematical expression:
![y - k = C \cdot (x-h)^{2}](https://tex.z-dn.net/?f=y%20-%20k%20%3D%20C%20%5Ccdot%20%28x-h%29%5E%7B2%7D)
Where:
- Horizontal component of the vertix, measured in inches.
- Vertical component of the vertix, measured in inches.
- Parabola constant, dimensionless. (Where vertix is an absolute maximum when
or an absolute minimum when
)
For the design of the door, the parabola must have an absolute maximum and x-intercepts must exist. The following information is required after considering symmetry:
(Vertix)
(x-Intercept)
(x-Intercept)
The following equation are constructed from the definition of a parabola:
![0-32 = C \cdot (28 - 0)^{2}](https://tex.z-dn.net/?f=0-32%20%3D%20C%20%5Ccdot%20%2828%20-%200%29%5E%7B2%7D)
![-32 = 784\cdot C](https://tex.z-dn.net/?f=-32%20%3D%20784%5Ccdot%20C)
![C = -\frac{2}{49}](https://tex.z-dn.net/?f=C%20%3D%20-%5Cfrac%7B2%7D%7B49%7D)
The parabolic shape of the door is represented by
. Now, the representation of the equation is included below as attachment.
At x = 0 inches and y = 22 inches, the distance from the edge of the door that head must observed to avoid being hit is:
![y -32 = -\frac{2}{49} \cdot x^{2}](https://tex.z-dn.net/?f=y%20-32%20%3D%20-%5Cfrac%7B2%7D%7B49%7D%20%5Ccdot%20x%5E%7B2%7D)
![x^{2} = -\frac{49}{2}\cdot (y-32)](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%3D%20-%5Cfrac%7B49%7D%7B2%7D%5Ccdot%20%28y-32%29)
![x = \sqrt{-\frac{49}{2}\cdot (y-32) }](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%7B-%5Cfrac%7B49%7D%7B2%7D%5Ccdot%20%28y-32%29%20%7D)
If y = 22 inches, then x is:
![x = \sqrt{-\frac{49}{2}\cdot (22-32)}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%7B-%5Cfrac%7B49%7D%7B2%7D%5Ccdot%20%2822-32%29%7D)
![x = \pm 7\sqrt{5}\,in](https://tex.z-dn.net/?f=x%20%3D%20%5Cpm%207%5Csqrt%7B5%7D%5C%2Cin)
![x \approx \pm 15.652\,in](https://tex.z-dn.net/?f=x%20%5Capprox%20%5Cpm%2015.652%5C%2Cin)
Head must 15.652 inches away from the edge of the door.