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Question 1091498: Suppose you are a Chief Mathematician for the A^2B^2 Company. Your company has a contract to build a football stadium in the form of two concentric ellipses, with the field inside the inner ellipse and seats between the two ellipses. The seats are in the intersection of the graphs pf x^2+4y^2 (greater than or equal sign) 100 and 25x^2+36y^2 (less than or equal sign) 3600 where each unit represents 10m. Draw the graph of the seating area.
In you research, you find out that the area of an elliptical region is (pi sign)ab, where a and b are half the lengths of the major and minor axes respectively. The Engineering Department estimates that each seat occuies 0.8m^2. What is the capacity of the stadium?
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! First put the equations for the two ellipses in standard form, (x/a)^2+(y/b)^2=1
The area comprising the seats is the intersection of the two regions bounded by the inner and outer ellipse
The inner ellipse in standard form is:
x^2 + 4y^2 >= 100 -> x^2/100 + 4y^2/100 >= 1 -> (x/10)^2 + (y/5)^2 >= 1
The outer ellipse in standard form is:
(25/3600)x^2 + (36/3600)y^2 <= 1 -> (x/12)^2 + (y/10)^2 <= 1
The difference in the two areas is:
pi*12*10 - pi*10*5 = pi*70 = 219.91, but since each unit represents 10 m, we multiply by 100 m^2 to get the area:
A = 21991 m^2
Since each set occupies 0.8 m^2, there are A/0.8 = 27488.94 which rounds down to 27488 seats
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