1. The U.S. Department of Energy monitors energy consumption in the United States. Petrolium consumption is of particular interest. In 1960, U.S. petrolium consumption was 20 quadrilion BTU's (British Thermal Units, a measure of energy). Petrolium consumption grew to 38 quadrillion BTU's in the year 2000. Assume that petrolium consumption is growing exponentially, and find an exponential growth model for the population of California. [Be sure to clearly define your variables to get started.] Then use your model to predict the petrolium consumption in the year 2010 and to predict when the petrolium consumption will reach 50 quadrillion BTU's.
model:
in 2010:
reach 50 quadrillion?
Answer
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 \(\boxed{2057}\).
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 \(\boxed{2057}\).
