There is a complex relation between the fuel consumption of a car and the speed of the car itself. For low speeds the engine cannot work optimally while for high speeds factors such as air resistance play an important role. In reality this means that the engine is least effective at very slow speeds and very high speeds, and the optimum is somewhere in between. Below is some data of velocities (km/h) and corresponding fuel consumed for 10km traveled (liters/10km). [velocity km/h] fuel 1/1 11/10km ] = [22 [10 25 40 55 70 90 105 120] 2.2 1.5 0.83 0.81 0.78 0.75 0.87 0.99 A common way to model fuel consumption is using a 3rd order polynomial. a) Use ordinary least squares to fit a 3rd polynomial to the data for fuel consumption as function of velocity. b) Rewrite the above data and use ordinary least squares to fit a 3rd order polynomial for fuel/hour as a function of velocity. c) Plot the original data points as well as the estimated fuel consumption (1/10km) for both models in the same figure for the interval 10-140km/h. d) Which of the two models have the largest residuals when used to estimate the fuel consumption (1/10km)? Which of the two models give a more realistic estimate of the fuel consumption (1/10km) when the car is going at 140km/h? Motivate your answers. A traveler is trying to decide between three different routes while traveling from city A to city D. The following relevant distances and speed limits exist: A B: 80 km/h speed limit, 60km distance. A C: 110 km/h speed limit, 100km distance. B C: 80 km/h speed limit, 50km distance. B D: 70 km/h speed limit, 100km distance. CD: 90 km/h speed limit, 40km distance. e) Assume the traveler is traveling at the speed limit. Based on your best model calculated earlier: which route should they use to go from A to D if they want to minimize the travel time? Which route if they want to minimize fuel consumption?