A rugby field is similar to a modern football field except that the goalpost, which is 18.5 ft wide, is located on the goal line instead of at the back of the endzone. The rugby equivalent of a touchdown, called a try, is scored by touching the ball down beyond the goal line. After a try is scored, the scoring team can earn extra points by kicking the ball through the goalposts. The ball must be placed somewhere on the line perpendicular to the goal line and passing through the point where the try was scored. See the figure below on the left.

If that line passes through the goalposts, then the kicker should place the ball at whatever distance is most comfortable. If the line passes outside the goalposts, then the player might choose the point on the line where angle u in the figure on the left is as large as possible. The problem of determining this optimal point is similar to a problem posed in 1471 by the astronomer Regiomontanus.

The figure on the right below shows a vertical line segment AB, where A and B are a and b units above the horizontal axis, respectively. If point P is located on the axis at a distance of x units from point Q, then angle u is greatest when x = √ab.

(a) Use the result from Regiomontanus’ problem to show that when the line is outside the goalposts, the optimal location to kick the rugby ball lies on the following hyperbola. x² - y² = 9.25²

(b) If the line on which the ball must be kicked is 10 ft to the right of the goalpost, how far from the goal line should the ball be placed to maximize angle u? Round to the nearest tenth.

(c) Rugby players find it easier to kick the ball from the hyperbola’s asymptote. When the line on which the ball must be kicked is 10 ft to the right of the goalpost, how far will this point differ from the exact optimal location? Round to the nearest tenth.

Transcribed Image Text:

The ball must be kicked from this line. 9.25 ft Goalposts Try scored here. B.