Question

In 1993, the sports league introduced a salary cap that limits the amount of money spent on players' salaries. The quadratic model y =0.2313x^2+2.600x+35.17 approximates this cap in millions of dollars for the years 1992​-2008​, where x=0 represents 1992​, x=1 represents 1993​, and so on. The approximate sports league salary cap in 2004 is $99.7 million.
Required:
In what year did the salary cap reach 65 million dollars?

Answer 1

Answer:

1999

Step-by-step explanation:

y  = 0.2313(y - 1992)^2 + 2.600(y - 1992) + 35.17

verify with known data point

y  = 0.2313(2004 - 1992)^2 + 2.600(2004 - 1992) + 35.17

y  = 0.2313(12)^2 + 2.600(12) + 35.17

y  = 27.9873 + 28.6 + 35.17

y = 99.6772  which verifies our equation

65 = 0.2313x² + 2.6x + 35.17

 0 = 0.2313x² + 2.6x - 29.83

quadratic formula

x = (-2.6 ±√(2.6² - 4(0.2313)(-29.83))) / (2(0.2313))

x = (-2.6 + 5.86) / 0.4626 = 7.05 years

7.05 = y - 1992

y = 1999.05