Mathematics: Ahighway tunnel has a shape that can be modelled by the equation of a parabola. the tunnel is 18m wide and the height of the tunnel 16m from the edge is 5m. can a truck 8m tall and 4m wide pass through the tunnel?

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Ahighway tunnel has a shape that can be modelled by

jaanne

05.07.2020

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)

Since the width of the tunnel is 18 m wide,

So, We consider the endpoints :

x_1=18,x_2=0

So, our equation becomes,

$y=a(x-18)(x-0)\\\\y=ax(x-18)$

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,

$y=a\times 16(16-18)\\\\5=a\times 16\times -2\\\\a=\frac{-5}{32}$

so, finally, equation becomes,

$y=\frac{-5}{32}x(x-18)$

Now, dimensions of truck is 8 m tall and 4 m wide.

So, y = 8

So,

$8=\frac{-5}{32}x(x-18)\\\\32\times 8=-5(x(x-18))\\\\256=-5x^2+90x\\\\5x^2-90x+256=0$

By using the "Quadratic formula " , we get:

$x_1=14.5\ and\ x_2=3.5$

So, Now, we get hte distance between the points x_1 and x_2

So, it becomes

$x_1-x_2=14.5-3.5\\\\=11\ m$

Allowed width is given by

$\frac{11}{2}=5.5\ m$

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.

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