Answer: It is possible that a truck of 8 m tall and 4 m wide pass through the tunnel.
Step-by-step explanation:
Since we have given that
A highway tunnel has a shape that can be modeled by equation of a parabola:
![y=a(x-x_1)(x-x_2)](https://tex.z-dn.net/?f=y%3Da%28x-x_1%29%28x-x_2%29)
Since the width of the tunnel is 18 m wide,
So, We consider the endpoints :
![x_1=18,x_2=0](https://tex.z-dn.net/?f=x_1%3D18%2Cx_2%3D0)
So, our equation becomes,
![y=a(x-18)(x-0)\\\\y=ax(x-18)](https://tex.z-dn.net/?f=y%3Da%28x-18%29%28x-0%29%5C%5C%5C%5Cy%3Dax%28x-18%29)
Now, we have given that the height of the tunnel is 16 m and the edge of tunnel is 5 m say 'y'.
so, it becomes,
![y=a\times 16(16-18)\\\\5=a\times 16\times -2\\\\a=\frac{-5}{32}](https://tex.z-dn.net/?f=y%3Da%5Ctimes%2016%2816-18%29%5C%5C%5C%5C5%3Da%5Ctimes%2016%5Ctimes%20-2%5C%5C%5C%5Ca%3D%5Cfrac%7B-5%7D%7B32%7D)
so, finally, equation becomes,
![y=\frac{-5}{32}x(x-18)](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-5%7D%7B32%7Dx%28x-18%29)
Now, dimensions of truck is 8 m tall and 4 m wide.
So, y = 8
So,
![8=\frac{-5}{32}x(x-18)\\\\32\times 8=-5(x(x-18))\\\\256=-5x^2+90x\\\\5x^2-90x+256=0](https://tex.z-dn.net/?f=8%3D%5Cfrac%7B-5%7D%7B32%7Dx%28x-18%29%5C%5C%5C%5C32%5Ctimes%208%3D-5%28x%28x-18%29%29%5C%5C%5C%5C256%3D-5x%5E2%2B90x%5C%5C%5C%5C5x%5E2-90x%2B256%3D0)
By using the "Quadratic formula " , we get:
![x_1=14.5\ and\ x_2=3.5](https://tex.z-dn.net/?f=x_1%3D14.5%5C%20and%5C%20x_2%3D3.5)
So, Now, we get hte distance between the points
and ![x_2](https://tex.z-dn.net/?f=x_2)
So, it becomes
![x_1-x_2=14.5-3.5\\\\=11\ m](https://tex.z-dn.net/?f=x_1-x_2%3D14.5-3.5%5C%5C%5C%5C%3D11%5C%20m)
Allowed width is given by
![\frac{11}{2}=5.5\ m](https://tex.z-dn.net/?f=%5Cfrac%7B11%7D%7B2%7D%3D5.5%5C%20m)
And the width of the truck is 4m
So , it is possible that a truck of 8 m tall and 4 m wide pass through the tunnel.