Based on data from 1999 to 2001, the rate of change in the annual net sales of Pepsi-Cola North America may be modeled by
R(t) = −131t − 749.5 million dollars per year
and the rate of change in the annual operating profit may be modeled by
P(t) = 12t + 76 million dollars per year, where t is the number of years since the end of 1999.
Determine the accumulated change in annual operating costs from the end of 1999 through 2001 by finding the area between these two curves.
To determine the accumulated change in annual operating costs from the end of 1999 through 2001, we need to find the area between the two given curves.
To find the area between two curves, we need to integrate the difference between the two functions over the desired interval.
First, let's find the difference between the two functions:
D(t) = P(t) - R(t)
= (12t + 76) - (-131t - 749.5)
= 12t + 131t + 76 + 749.5
= 143t + 825.5
Now, we need to integrate D(t) from the end of 1999 (t = 0) through 2001 (t = 2):
∫[0,2] (143t + 825.5) dt
To integrate this, we can use the power rule of integration:
∫(at + b) dt = (1/2)at^2 + bt + C
Applying this rule to our integral:
∫[0,2] (143t + 825.5) dt = (1/2)(143)t^2 + (825.5)t + C
Evaluating the definite integral from 0 to 2:
[(1/2)(143)(2)^2 + (825.5)(2)] - [(1/2)(143)(0)^2 + (825.5)(0)]
= (1/2)(143)(4) + (825.5)(2)
= 286 + 1651
= 1937 million dollars
Therefore, the accumulated change in annual operating costs from the end of 1999 through 2001 is 1937 million dollars.