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An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

2.   An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

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Since the Axis of the parabola is vertical, Let the equation of the parabola be,

x^2=4ay

 it can be seen that this curve will pass through the point (5/2, 10) if we assume origin at the bottom end of the parabolic arch.

So,

\left(\frac{5}{2}\right)^2=4a(10)

a=\frac{25}{160}=\frac{5}{32}

Hence, the equation of the parabola is 

x^2=4\times\frac{5}{32}\times y

x^2=\frac{20}{32} y

x^2=\frac{5}{8} y

Now, when y = 2 the value of x will be 

x=\sqrt{(\frac{5}{8}\times2)}=\sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2}

Hence the width of the arch at this height is 

2x=2\times\frac{\sqrt{5}}{2}=\sqrt{5}.

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