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 find the accumulated change in annual operating costs from the end of 1999 through 2001, we need to find the area between the curves of R(t) and P(t) over the interval [0, 2].
The accumulated change in costs can be determined by calculating the definite integral of the difference between R(t) and P(t) over the given interval.
First, let's find the difference between the two functions:
R(t) - P(t) = (-131t - 749.5) - (12t + 76)
= -131t - 749.5 - 12t - 76
= -143t - 825.5
Now we can find the definite integral of -143t - 825.5 with respect to t over the interval [0, 2]:
∫[-143t - 825.5]dt from 0 to 2
To calculate this integral, we can use the power rule of integration.
∫[-143t - 825.5]dt = -71.5t^2 - 825.5t + C
Now we evaluate the integral between the limits 0 and 2:
[-71.5(2)^2 - 825.5(2)] - [-71.5(0)^2 - 825.5(0)]
[-71.5(4) - 825.5(2)] - [0 - 0]
[-286 - 1651] - [0]
-1937 - 0
-1937 million dollars
Therefore, the accumulated change in annual operating costs from the end of 1999 through 2001 is -1937 million dollars.