Subject: applications of quadratic relations

I can't figure this question out:

A tunnel is built under a river for a road 12m wide with a 2m sidewalk on either side. The top of the tunnel is semi-elliptical. A local bylaw stipulates that there must be a clearance of at least 3.6m at all points on the road. If the smallest possible ellipse is used, find the clearance at the center of the road. Thanks!

Hi Janna,

Put a coordinate system on the plane with the center of the ellipse at the origin and the road on the x-axis, then the equation of the ellipse is

 x2/a2 + y2/b2 = 1 Since the ellipse passes through (8,0) you know that  82/a2 = 1 and hence a = 8.

You also know that the ellipse passes through (6,3.6) and thus

 62/82 + 3.62/b2 = 1 Solve for b.

Cheers,
Harley
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