Question

railroad tunnel is shaped like a parabola:the height of the tunnel at the center is 54 ft and the vertical clearance must be 18 ft at a point 8 ft from the center.find an equation for the parabola

Answer 1

We have a parabola.

We know that, at the center, x=0, the height is 54.

So we have the vertex of the parabola at (0,54).

`Vertex\colon(h,k)=(0,54)`

As the clearance at height y=8 is 18, we know that for points x=-18/2=-9 and x=18/2=9, the value of y is y=8.

We have two points of the parabola:

`\begin{gathered} (x,y)=(-9,8) \\ (x,y)=(9,8) \end{gathered}`

We can use the vertex form to describe the parabola:

`\begin{gathered} \text{Vertex form}\longrightarrow y=a(x-h)^2+k \\ y=a(x-0)^2+54=ax^2+54 \end{gathered}`

We use one of the points to find the value of the parameter a:

`\begin{gathered} (x,y)=(9,8) \\ 8=a\cdot9^2+54 \\ 8-54=81a \\ -46=81a \\ a=-(46)/(81) \end{gathered}`

Then, the equation becomes:

`y=-(46)/(81)x^2+54`

Answer: y=-46/81*x^2+54

We can graph this as: