Final Answer:
(a) The flow rate equation, \(R(t)\), for Sara Lee Corporation's revenue growth, increasing by 5% per year, is expressed as \(R(t) = 12.88 \times (1 + 0.05)^t\) billion dollars per year.
(b) The future value of the investment at the end of December 31, 2012, with a continuous investment of 3.5% of the revenue at an APR of 5%, amounts to approximately $0.601 billion.
Step-by-step explanation:
(a) The flow rate equation, \(R(t)\), describes Sara Lee's revenue growth over time. Given that the initial revenue in 2009 was $12.88 billion and it increases by 5% annually, the equation becomes \(R(t) = 12.88 \times (1 + 0.05)^t\), where \(t\) represents the number of years after July 1, 2009. This equation models the revenue in billions of dollars per year.
(b) To find the future value of the investment made annually, compounded continuously at an APR of 5%, the formula for continuous compounding \(A = P \times e^{rt}\) is utilized. Here, the continuous investment is 3.5% of the annual revenue, or \(0.035 \times R(t)\), with \(r = 0.05\) (the APR) and \(t = 3\) (for the 3.5 years from July 1, 2009, to December 31, 2012). Substituting these values into the formula, the future value of the investment at the end of December 31, 2012, is calculated to be approximately $0.601 billion.
Understanding and applying the flow rate equation for revenue growth and utilizing the continuous compounding formula helps compute the future value of Sara Lee's investment based on the specified conditions for revenue increase and continuous investment.