Artillerymen on a hillside are trying to hit a target behind a mountain on the other side of a river. Their cannon is at (x, y) = (3, 250) where x is in kilometers and y is in meters. The target is at (x, y) = (-2, 50). In order to avoid hitting the mountain on the other side of the river, the projectile from the cannon must go through the point (x, y) = (-1, 410).

Write the equation for the problem.


The physical principle that makes this possible is that if you neglect reststance then the path of the cannonball is a parabola. Suppose the parabola is

y = a x2 + b x + c

Since the points (3,250), (-2,50) and (-1,410) are on the path of the cannonball these points must satisfy the equation. That is

250 = 32 a + 3 b + c
50 = (-2)2 a - 2 b + c and
410 = (-1)2 a - b + c

If you solve these three equations for a, b and c you will have the equation that descrives the path of the cannonball.

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