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 two curves R(t) and P(t).

First, let's find the two values of t that correspond to the years 1999 and 2001. Since t represents the number of years since the end of 1999, we subtract 1999 from each given year:

For the end of 1999: t = 1999 - 1999 = 0.
For the end of 2001: t = 2001 - 1999 = 2.

Now, we can calculate the accumulated change in annual operating costs by finding the area between the two curves.

The equation R(t) = −131t − 749.5 represents the annual operating costs, and the equation P(t) = 12t + 76 represents the rate of change in the annual operating profit.

To find the area between these two curves, we need to subtract the integral of P(t) from the integral of R(t) over the interval [0, 2].

The integral of R(t) with respect to t is:
∫(−131t − 749.5) dt = -131(t^2/2) - 749.5t + C1, where C1 is the constant of integration.

The integral of P(t) with respect to t is:
∫(12t + 76) dt = 6t^2 + 76t + C2, where C2 is the constant of integration.

Now, we can substitute the values of t = 0 and t = 2 into the two integrals:

Area = ∫(−131t − 749.5) dt - ∫(12t + 76) dt
= [-131(t^2/2) - 749.5t + C1] - [6t^2 + 76t + C2]
= -131(2^2/2) - 749.5(2) + C1 - (6(2^2) + 76(2) + C2)
= -131(4/2) - 749.5(2) + C1 - (6(4) + 76(2) + C2)
= -131(2) - 749.5(2) - 6(4) - 76(2) + C1 - C2
= -262 - 1499 - 24 - 152 + C1 - C2
= -2937 + C1 - C2.

So, the accumulated change in annual operating costs from the end of 1999 through 2001 is -2937 + C1 - C2 million dollars per year.