In 1996, the sports league introduced a salary cap that limits the amount of money spent on players' salaries. The quadratic model $y=0.2313 x^{2}+2.600 x+35.17$ approximates this cap in millions of dollars for the years $1996-2006$, where $\mathrm{x}=0$ represents $1996, \mathrm{x}=1$ represents 1997 , and so on. Complete parts $\mathrm{a}$ and $\mathrm{b}$.
a. Approximate the sports league salary cap in 2004.
The approximate sports league salary cap in 2004 is $\$ 70.8$ million. (Round to the nearest tenth as needed.)
b. According to the model, in what year did the salary cap reach 70 million dollars?
According to the model, in the salary cap reached 70 million dollars. (Round down to the nearest year.)
Answer
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 \(\boxed{2003}\).
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 \(\boxed{2003}\).
