Projected methane emissions in the US can be modeled by fx 0

Projected methane emissions in the U.S. can be modeled by f(x) =-0.072x^2 + 1.93x + 173.9 where f (x) is the amount of methane produced in million metric tons after the year 2000 a. According to this model what will U.S. emissions of methane be in 2018? h. In what year will the emissions in the U.S. be at their maximum? c. What is the level of methane emissions for the year in part b?

Solution

f(x) = -0.072x² + 1.93x + 173.9

year 2000 ==> x = 0

==> year 2018 ==> x = 18

a) x = 18 ==> f(x) = -0.072(18)² + 1.93(18) + 173.9

==> f(x) = 185.312

Hence U.S emissions of methane in 2018 is 185.312 million metric tons

b) maximum emissions ==> f \'(x) = 0

==> -0.072(2)x^2-1 + 1.93(1) + 0 = 0

==> -0.144x + 1.93 = 0

==> x = 1.93/0.144

==> x = 13.403

==> x = 13

x = 13 ==> year 2013

Hence U.S emissions of methane are maximum in the year 2013

c) at x = 13 , f(x) = -0.072(13)² + 1.93(13) + 173.9

==> f(x) = 186.822

Hence emissions of the year 2013 are 186.822 million metric tons