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. Graph the two equations on the same set of coordinate axes.
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 represented by the given equations.
The equation for the rate of change in annual net sales is given as:
R(t) = -131t - 749.5 million dollars per year
And the equation for the rate of change in annual operating profit is given as:
P(t) = 12t + 76 million dollars per year
To find the area between the two curves, we need to integrate the difference between the two functions over the interval from t-value representing the end of 1999 to the t-value representing the end of 2001.
Let's set up the integral:
∫[t0 to t1] (P(t) - R(t)) dt
Now, let's find the values for t0 and t1:
The end of 1999 is the starting point, so t0 = 0.
The end of 2001 can be calculated by subtracting the number of years from the end of 2001 to the end of 1999. Since the end of 1999 was the starting point, the number of years is 2001 - 1999 = 2. Therefore, t1 = 2.
Now, substitute the given equations into the integral:
∫[0 to 2] ((12t + 76) - (-131t - 749.5)) dt
Simplify the integrand:
∫[0 to 2] (143t + 825.5) dt
Integrate the function:
∫[0 to 2] (143t + 825.5) dt = [71.5t^2 + 825.5t] evaluated from 0 to 2
Evaluate the integral at the upper limit (2) and lower limit (0):
[71.5(2)^2 + 825.5(2)] - [71.5(0)^2 + 825.5(0)]
Simplify the expression:
[143(4) + 1651] - [0 + 0]
= 572 + 1651
= 2223 million dollars per year
Therefore, the accumulated change in annual operating costs from the end of 1999 through 2001 is 2223 million dollars per year.
To graph the two equations on the same set of coordinate axes, plot the points for each equation and connect them with a line:
For R(t) = -131t - 749.5, you can choose any two t-values, substitute them into the equation, and plot the resulting points. Then connect the points with a line.
For example, when t = 0, R(t) = -749.5, so plot (0, -749.5). When t = 2, R(t) = -131(2) - 749.5 = -1011.5, so plot (2, -1011.5). Connect the two points with a line.
For P(t) = 12t + 76, follow the same process. For example, when t = 0, P(t) = 76, so plot (0, 76). When t = 2, P(t) = 12(2) + 76 = 100, so plot (2, 100). Connect the two points with a line.
Now, draw the x-axis representing the time (t) and the y-axis representing the amount in million dollars per year. Plot the points and connect them with lines for both equations.