Chapter 1.1, Problem 62E

Oil ProductionThe following table shows the amount of crude oil in billions of barrels produced in the United States in recent years. Source: U.S. Energy Information Administration. Year Crude Oil Produced 2002 2.097 2003 2.060 2004 1.989 2005 1.893 2006 1.857 2007 1.853 2008 1.830 2009 1.954 2010 2.000 2011 2.063 2012 2.377 In this exercise we are interested in the total amount of crude oil produced over the 10-year period from mid-2002 to mid-2012, using the data for the 11 years above. One approach is to sum up the numbers in the second column, but only count half of the first and last numbers. Give the answer to this calculation. Approximate the amount of crude oil produced over the 10-year period 2002-2012 by taking the average of the left endpoint sum and the right endpoint sum. Explain why this is equivalent to the calculation done in part a. This is also equivalent to a formula known as the trapezoidal rule, discussed in the next chapter. If your calculator has a cubic regression feature, find the best-fitting cubic function for these data, letting t=0 correspond to 2000. Then integrate this equation over the interval [2.12] to estimate the amount of crude oil produced over this time period. Compare with your answer to part a.

Civil War Estimates of the Confederate Armys strength and losses for the battle of Gettysburg are given in the following table, where strength is the number of soldiers immediately preceding the battle and loss indicates a soldier who was killed, wounded, captured, or missing. Source: Regimental Strengths and Losses at Gettysburg. Unit Strength Loss I Corps Longstreet 20,766 7661 II Corps Ewell 20,666 6603 III Corps Hill 22,083 8807 Cavalry Stuart 6621 286 Total 70,076 22,557 a. Find the probability that a randomly selected confederate soldier was from the III Corps. b. Find the probability that a confederate soldier was lost in the battle. c. Find the probability that a I Corps soldier was lost in the battle. d. Which group had the highest probability of not being lost in the battle? e. Which group had the highest probability of loss?

Bronze is an alloy of copper, zinc, and tin with small amounts of other elements. Two types of bronze castings are listed in the table below with the percent composition of copper, tin, and zinc in each casting. Determine the ratios called for in the table.