Question

In 1993, the sports league introduced a salary cap that limits the amount of money spent on players' salaries. The quadratic model y =0.2313x2 +2.600x + 35.17

approximates this cap in millions of dollars for the years 1993-2013, where x = 0 represents 1993, X = 1 represents 1994, and so on. Complete parts a and b.
a. Approximate the sports league salary cap in 2009.
me
nts
ontents
The approximate sports league salary cap in 2009 is $ million
(Round to the nearest tenth as needed.)
b. According to the model, in what year did the salary cap reach 65 million dollars?
Cuccess
According to the model in the salary cap reached 65 million dollars.
(Round down to the nearest year)
ts for a

Answer 1

a. The approximated salary cap in 2009 is $136.0 millions

b. The salary cap reached 65 million dollars in 2000

Explanation:

* Lets explain how to solve the problem

- The quadratic model of the salary cap in million is

y = 0.2313 x² + 2.600 x + 35.17

- The approximation of this cap in millions of dollars for the years

1993-2013 where x = 0 represents 1993, x = 1 represents 1994,

and so on

a. Lets calculate the approximated sports league salary cap in 2009

∵ x at 2009 = 2009 - 1993 = 16

∵ y = 0.2313 x² + 2.600 x + 35.17

∴ y = 0.2313 (16)² + 2.600 (16) + 35.17

∴ y = 135.98 ≅ 136.0 millions

* The approximated salary cap in 2009 is $136.0 millions

b. Lets calculate in what year did the salary cap reach 65 million dollars

∵ y = 65

∵ y = 0.2313 x² + 2.600 x + 35.17

∴ 65 = 0.2313 x² + 2.600 x + 35.17

- Subtract 65 from both sides

∴ 0.2313 x² + 2.600 x - 29.83 = 0

- Use the calculator to find the value of x by solving the quadratic

equation

∴ x = 7.05 and x = -18.29 (we will reject this value)

∴ x ≅ 7 years

∴ The salary cap reached 65 million dollars in (1993 + 7) = 2000

* The salary cap reached 65 million dollars in 2000