The physicist Galileo observed that certain projectiles follow a parabolic path. For instance, if a cannon fires a shell at a 45° angle with a speed of v feet per second, then the path of the shell (see the figure on the left below) is modeled by the following equation.

The figure on the right below shows the paths of shells all fired at the same speed but at different angles. The greatest distance is achieved with a 45° angle. The outline, or envelope, of this family of curves is another parabola with the cannon as focus. The horizontal line through the vertex of the envelope parabola is a directrix for all the other parabolas. Suppose all the shells are fired at a speed of 252.982 ft per sec.

(a) What is the greatest distance, to the nearest foot, that a shell can be fired?

(b) What is the equation of the envelope parabola?

(c) Can a shell reach a helicopter 1500 ft due east of the cannon flying at a height of 450 ft?

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