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Question

A tunnel through a mountain for a four-lane highway is to have an elliptical opening.

The total width of the highway (not the opening) is to be 16m, and the height at the edge of the road must be sufficient for a truck 4m high to clear if the highest point of the opening is to be 5m approximately.

How wide must the opening be,


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Solution

Calculate the width of the opening of the tunnel.

A diagram is shown below which describes the situation of the question:

in the diagram, A'A represent the width of the opening of the tunnel, B'B represents the width of the highway.

Given that, B'B=16mOB=8m

Since the shape of the tunnel is like an ellipse.

The equation of an ellipse is given as,

x2a2+y2b2=1

Where 2a represent the length of the major axis and 2b represent the length of the minor axis.

Given that, the highest point of the opening is to be 5m approximately.
So, this 5m will be the total height which is the semi-minor axis of the ellipse.

b=5m

From the figure, it is observed that the point (8,4) lies on the ellipse.

82a2+4252=164a2+1625=164a2=1-162564a2=925a2=25×649a2=16009a=16009a=403a=13.34m

The width of the opening of the tunnel is computed as follow:

A'A=2a=2×13.34=26.68m

Hence, the width of the opening of the tunnel is 26.68m.


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