Step 1 :Given the quadratic model \(y=0.2313 x^{2}+2.600 x+35.17\) which approximates the sports league salary cap in millions of dollars for the years 1996-2006, where \(x=0\) represents 1996, \(x=1\) represents 1997, and so on.
Step 2 :To approximate the sports league salary cap in 2004, we need to substitute \(x=8\) (2004 - 1996) into the quadratic model and calculate the value of \(y\).
Step 3 :The approximate sports league salary cap in 2004 is \(y = 0.2313*8^{2} + 2.6*8 + 35.17 = 70.7732000000000\) million dollars. Rounding to the nearest tenth, we get \(70.8\) million dollars.
Step 4 :To find the year when the salary cap reached 70 million dollars, we need to set \(y=70\) in the quadratic model and solve for \(x\).
Step 5 :Solving the equation \(70 = 0.2313*x^{2} + 2.6*x + 35.17\) gives two solutions: \(x = -19.1175403785827\) and \(x = 7.87672758134968\).
Step 6 :Since \(x\) represents the years after 1996, a negative value doesn't make sense in this context. So, we only consider the positive solution.
Step 7 :The value of \(x=7.87672758134968\) corresponds to the year 2004 (1996 + 7.87672758134968 = 2003.87672758134968). Since we need to round down to the nearest year, the salary cap reached 70 million dollars in the year 2003.
Step 8 :Final Answer: a. The approximate sports league salary cap in 2004 is \(\boxed{70.8}\) million dollars. b. According to the model, the salary cap reached 70 million dollars in the year \(\boxed{2003}\).