Question

A tunnel is in the shape of a parabola. The maximum height is 27 m and it is 19 m wide at the base.
What is the vertical clearance 8 m from the edge of the tunnel?

Answer 1

Adn: We need to model that inverted vertical parabola with a quadratic equation.
The simplest form of a vertical parabola is y=x^2. This has its vertex at (0,0), as does the parabola in the illustration for this problem.

Inserting a constant coefficient makes the equation more general: y=ax^2.
Inverting the graph (so that the parabola opens downward) results in y=-ax^2.

If the base of the tunnel is 19 meters, half of that is 9.5 meters. Please satisfy yourself that if x=9.5 meters in the illustration, then the the y value is -27 (since the tunnel is 27 meters high at its peak).

We then have enough info to find the value of a: y=-ax^2 => -27=-a(9.5)^2.

Please find a.

Then write the equation of your inverted parabola by substituting your value for a into y=-ax^2.
Now, to predict the height of the tunnel when x=8 meters, simply substitute -8 for x and find y.