MATH 1113 PreCalculus

Day 31, Project #4
Maximum Angle

On a football field, whether high school, college or professional, when a play begins, depending on the result of the previous play, the ball may be set in the center of the field, or on one of the "hash marks," running down the left and right sides of the field.

In the case where the ball is placed on a hash mark, if a kicker is kicking a field goal, there's a bit of offset to the angle in attempting to get the ball between the goalpost uprights.

Here's a picture of a kicker kicking from a hash mark:

http://www.vsbw.com/~terry/Rice/MVC-489X.JPG

Your project is to use inverse trig functions to determine how far away from the goal posts the ball should be placed in order to maximize the angle the kicker has available to get the ball between the uprights.

To see what's going on, look at the following Geogebra applet.  Notice that you'll have to set the field up according to official NCAA dimensions.  Try a Google search using something like "NCAA football field dimensions" and you should find the relevant values (there's a picture available at ncaa.org for sure.)  In the applet, once the goalpost width and hash marks are properly placed (using the sliders), you can move the position of the ball forwards and backwards and observe the resulting angle:

Applet

Here's your assignment:

1.  Find the correct values for the "distance from center of field to inner edge of goal post," and "distance from center of field to center of hash mark -- notice that the hash marks are 2 ft. wide, so to get to the center, you have to add a foot.

2. Although the actual goalposts are 10 ft. above the ground, we're going to drop everything down to the same level, just to make it easier to work with (it turns out that you can "tip" the plane up, then project back down, and the actual values are nearly the same.) Once you've set the field up correctly, slide the ball position around and find the approximate angles associated with:

A) a 40 yard field goal (notice that you'll need to be out 40*3 = 120 feet from the BACK of the end zone, so you need to be 120 - 30 = 90 feet from the end zone on the Geogebra field.)

B) a 25 yard field goal (use the same logic as in part "A.")

3. Slide the ball position back and forth to locate the apparent maximum angle.  Describe this location.

4. Now for some trigonometry.  The angle, alpha, can be thought of as the difference between two angles associated with right triangles (the ones with hypotenuses attached to the goal posts.)  Express the angle as a function of the distance, x, from the back of the end zone (you'll need two Arctangent expressions.)

5.  Graph the function from problem #4, and locate the maximum value.  The answer you get should be about the same as your answer for problem #3. 

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Page Last Modified: 21 March, 2007

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