Width 50 height 30 sail with mast of 27

Question

The underside of a bridge has the shape of a parabolic arch. It has a maximum height of 30m and a width of 50m. Can a sailboat with a mast of 27m above the water pass under the bridge at a distance of 8m from the axis of symmetry of the arch?

Answer 1

The answer is no. The height of the bridge at that point is 26.928 m!

 

You can work this problem by finding the forumula for the arch wrt some origin. It is easiest to choose the origin to be at the point in the middle of the arch, at the water's surface. By inference, using the known formats for parabolas, you can infer that the function will be of the form:

 

y(x)= ax^2 + 30

 

where a is a constant to be determined by using the known information that y(25)= 0. This results in a value for a of -6/125 (if I did that right).

 

This gives a complete equation for the arch. To find out the height of the arch at x= 8 meters, just calculate

 

y(8)= (-6/125)(64) + 30= 26.928 (meters)

 

This gives the y-value of the parabola where x= 8, which is the height of the arch.