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 Question from Marioneta, a student: Two same columns placed symmetrically, 40 m apart. The height of columns is 8 m. If the origin of the coordinate system is placed at the foot of the left column the equation of the arc of the bridge: f (x) =-1/80x^2+1/2x+8 What is the maximum height of a boat sailing under the bridge and identify its path.

Marioneta,

By the form of the function $f(x) = - \frac{1}{80} x^2 + \frac12 x + 8$ the shape of the arc is a parabola opening downwards. When $x = 0$ m the height of the parabola is 8 m and when $x = 40$ m the height is again 8 m. You should check this. The vertex of the parabola is then midway between 0 m and 40 m. What is the height of the parabola there?

If the boat has a tall mast like a sailing ship then, if it sails down the centre of the channel it can have a height that just fits under the vertex. If it is box shaped then the maximum height will depend on the width of the boat.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.