The arc of a bridge - Math Central

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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

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