# 2007 Cyprus MO/Lyceum/Problem 23

## Problem

In the figure above the right section of a parabolic tunnel is presented. Its maximum height is $OC=8\,\mathrm{m}$ and its maximum width is $AB=20\,\mathrm{m}$. If M is the midpoint of $OB$, then the height $MK$ of the tunnel at the point $M$ is

$\mathrm{(A) \ }5\,\mathrm{m}\qquad \mathrm{(B) \ } 5.2\,\mathrm{m}\qquad \mathrm{(C) \ } 5.5\,\mathrm{m}\qquad \mathrm{(D) \ } 5.8\,\mathrm{m}\qquad \mathrm{(E) \ } 6\,\mathrm{m}$

## Solution

Since it is a parabolic tunnel, the equation of the tunnel is a quadratic. We have three points: (0,8), (10,0), and (-10,0). Since we have both of the roots, we multiply $a(x-10)(x+10)=ax^2-100a$. But we also have $-100a=8$, so $a=.08$. Thus the equation of the parabola is $-.08x^2+8$. Now the height of the tunnel at M is the value of the y coordinate when $x=5$, or $6$. $\mathrm{(E)}$