Question

A tunnel is in the shape of a parabola. The maximum height is 50 m and it is 10 m wide at the base, as shown below. A parabola opening down with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 50 meters and its width from left to right is 10 meters. What is the vertical clearance 2 m from the edge of the tunnel?

Answer 1

The 3 known points of the parabola are the origin (0,0) and the bottom left and right points ( -5,-50) (5,-50)

The general form of a parabola that goes through the origin is:  

y = ax^2  

We need to solve for a:

-50 = a(5)^2  

-50 = 25a  

a = -50/25

a = -2

Now the equation of the parabola is y = (-2)x^2.

To solve for the vertical clearance (y) 2 meters from the edge

Subtract the distance from the edge 2 meters from the original edge which is 5 meters, 5-2 = 3.

Now replace x with 3 and solve for y:

y = (-2)3^2

y = -2(9)

y = -18

The clearance would then 50 - 18 = 32 meters.