Answer:
117 gallons
Step-by-step explanation:
The expression that describes the rate of U.S. per capita sales of bottled water for the period 2007–2014 is:
[tex]s(t) = 0.25t^2-t+29[/tex]
Since t=0 at 2007. At 2009 and 2013, t is:
[tex]t_1=2009-2007 = 2\\t_2=2013-2007 = 6[/tex]
The definite integral of the expression from t=2 to t=6 is:
[tex]s(t) = 0.25t^2-t+29\\\int\limits^6_2 {s(t)} \, dt= \int\limits^6_2 {(0.25t^2-t+29)} \, dt\\\int\limits^6_2 {s(t)} \,dt= (\frac{t^3}{12} -\frac{t^2}{2}+29t +c)|^6_2 \\S(6) -S(2) = (\frac{6^3}{12} -\frac{6^2}{2}+29*6 +c) - (\frac{2^3}{12} -\frac{2^2}{2}+29*2 +c)\\S(6) -S(2) = 117.333\ gallons[/tex]
Rounded to the nearest gallon, the estimated total U.S. per capita sales of bottled water from the start of 2009 to the start of 2013 is 117 gallons.
