Step 1 :Define the variables for the problem. Let \(P(t)\) be the petroleum consumption at time \(t\), \(P_0\) be the initial petroleum consumption, \(k\) be the growth rate, and \(t\) be the time in years. In this case, \(P_0 = 20\) quadrillion BTU's, \(P(40) = 38\) quadrillion BTU's, \(t_0 = 0\), and \(t_{40} = 40\).
Step 2 :Use the given points to solve for the growth rate \(k\) in the exponential growth model \(P(t) = P_0 * e^{kt}\).
Step 3 :Substitute the given values into the equation to get \(38 = 20 * e^{40k}\). Solve for \(k\) to get \(k = 0.016\).
Step 4 :Use the growth rate to predict the petroleum consumption in 2010. Substitute \(t = 50\) and \(k = 0.016\) into the equation to get \(P(50) = 20 * e^{0.016 * 50}\), which simplifies to \(P(50) = 44.61\) quadrillion BTU's.
Step 5 :Use the growth rate to predict when the petroleum consumption will reach 50 quadrillion BTU's. Set \(P(t) = 50\) and solve for \(t\) to get \(t = \frac{\ln(50/20)}{0.016}\), which simplifies to \(t = 57.10\) years.
Step 6 :Since the initial year was 1960, add 57.10 to 1960 to get the year 2017.
Step 7 :Final Answer: The growth rate of petroleum consumption is approximately \(\boxed{0.016}\) per year. The predicted petroleum consumption in 2010 is approximately \(\boxed{44.61}\) quadrillion BTU's. The petroleum consumption is predicted to reach 50 quadrillion BTU's around the year \(\boxed{2057}\).