Baseball Field Area Problem
The South City Bombers are a pro baseball team that plays at Symmetry Field. The field is perfectly symmetrical across the line from home plate (0,0) to the center field wall, which is 400 ft away from home. The distance down the left and right field lines is 340 ft (and the two lines are perpendicular at home plate). Left field, center field, and right field are each divided evenly by the angle out from home plate. The infield portion of the diamond is cut by a perfect arc with a radius of 150 ft (the arc is a 90 degree arc). The outfield also has a 15 ft wide dirt warning track before the wall, and the total area of the track is 8,241.17 square ft. The owner of the team wants to host a Metallica concert at Symmetry Field, and knowing the patrons will be rough on the field, wants to negotiate for the band to pay for the cost of re-sodding the field. With negotiations going poorly, the owner decides he can settle for just having the outfield grass in fair territory (the area pictured) re-sodded. The sod company charges $1 per square foot, and Metallica refuses to pay one cent over $100,000 for the field. Do the two sides have a deal?