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:
[tex]y=a(x-x_1)(x-x_2)[/tex]
Since the width of the tunnel is 18 m wide,
So, We consider the endpoints :
[tex]x_1=18,x_2=0[/tex]
So, our equation becomes,
[tex]y=a(x-18)(x-0)\\\\y=ax(x-18)[/tex]
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,
[tex]y=a\times 16(16-18)\\\\5=a\times 16\times -2\\\\a=\frac{-5}{32}[/tex]
so, finally, equation becomes,
[tex]y=\frac{-5}{32}x(x-18)[/tex]
Now, dimensions of truck is 8 m tall and 4 m wide.
So, y = 8
So,
[tex]8=\frac{-5}{32}x(x-18)\\\\32\times 8=-5(x(x-18))\\\\256=-5x^2+90x\\\\5x^2-90x+256=0[/tex]
By using the "Quadratic formula " , we get:
[tex]x_1=14.5\ and\ x_2=3.5[/tex]
So, Now, we get hte distance between the points [tex]x_1[/tex] and [tex]x_2[/tex]
So, it becomes
[tex]x_1-x_2=14.5-3.5\\\\=11\ m[/tex]
Allowed width is given by
[tex]\frac{11}{2}=5.5\ m[/tex]
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.
