Australian football is played on an elliptical field. The official rules state
that the field must be between 135 and 185 meters long and between 110 and 155
meters wide.
a = semi-major axis's length
2a = major axis's length = length of field
>>...the field must be between 135 and 185 meters long...<<
135 < 2a < 185
Divide through by 2
67.5 < a < 92.5
b = semi-minor axis's length
2b = minor axis's width = width of field
>>...the field must be...between 110 and 155 meters wide...<<
110 < 2b < 155
Divide through by 2:
55 < b < 77.5
a) write an equation for the largest allowable playing field.
The formula for the area of an ellipse is
Area = pab
The area is latgest when a and b are the largest, that is when
a = 92.5 and b = 77.5
Largest area = p(92.5)(77.5)
= 22521.3 mē approximately.
b) write an equation for the smallest allowable playing field.
The area is smallest when a and b are the smallest, that is when
a = 67.5 and b = 55
Smallest area = p(67.5)(55)
= 11663.2 mē approximately.
c) write an inequality that describes the possible areas of an Australian
football field.
11663.2 mē < Area < 22521.3 mē
Edwin
AnlytcPhil@aol.com