Q:

# A tunnel is in the shape of a parabola. The maximum height is 16 m and it is 16 m wide at the base, as shown below. What is the vertical clearance 7 m from the edge of the tunnel?

Accepted Solution

A:

Explanation:

1) Since the vertex is at the origin, i.e point (0,0), and the simmetry axis is paralell to the vertical axis, you know that the canonical equation of the parabola is:

y = -4px², where p is the focal distance.

You can also use y = ax², where a = -4p.

2) Using y = ax², you can find a from the fact that at x = 8 (one edge of the tunnel), y = - 16 ( negative value of the height).

Substitute those values (8,16) in the equation y = ax², and you get:

- 16 = a (8²) ⇒ a = - 16 / 64 = - 1/4

Therefore, the equation of the parabola is: y = (-1/4)x².

3) To find the vertical clearance at 7 m from the edge of the tunnel, you must realize that the x-coordinate is x = 8 - 7 = 1.

Then, find y for x = 1:

y = (-1/4)(1²) = - 1/4 = - 0.25

That is the y-coordinate; the clearance is the 16m - 0.25m = 15.75m