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 curves representing the rate of change in the annual net sales and the rate of change in the annual operating profit.

To find the area between two curves, we need to find the integral of their difference. In this case, the difference between the two curves is:

R(t) - P(t) = (-131t - 749.5) - (12t + 76)

Simplifying this expression, we get:

R(t) - P(t) = -131t - 749.5 - 12t -76

Combining like terms, we have:

R(t) - P(t) = -143t - 825.5

Now we can find the integral of this expression to determine the accumulated change in annual operating costs over the given time period.

∫(-143t - 825.5) dt

To find the integral, we can apply the power rule of integration, which states that:

∫x^n dx = (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

Applying this rule to our expression, we get:

∫(-143t - 825.5) dt = (-143/2) * t^2 - 825.5t + C

Now, we need to evaluate this expression at the limits of integration, which are the years from the end of 1999 through 2001.

Since t represents the number of years since the end of 1999, for the year 2001, t = 2. For the end of 1999, t = 0.

Plugging these values into our expression, we get:

((-143/2) * 2^2 - 825.5 * 2) - ((-143/2) * 0^2 - 825.5 * 0)
= (-143/2) * 4 - 825.5 * 2
= -572/2 - 1651
= -286 - 1651
= -1937 million dollars

Therefore, the accumulated change in annual operating costs from the end of 1999 through 2001 is -1937 million dollars.