1. A common metric to evaluate the performance of a model of a car is its "O to 60" time. That is, the amount of time it takes to go from rest to 60 miles per hour. Recently, the Porsche 918 Spyder hybrid supercar and Tesla Model S both managed times of 2.2 seconds. However, most of our vehicles do not have these capabilities. Dr. Lawrence's 2011 Chevrolet Impala is rated as having a 0 to 60 time of roughly 8 seconds. Instead of working with miles per hour, which cars do not need an entire mile to reach 60 miles per hour nor do they need an hour of time to reach that speed, converting to feet per second makes more sense as we know multiple feet will be traveled and it will be done in a matter of seconds. (Think of traveling on highway on-ramps.) Quite nicely, converting 60 miles per hour is exactly 88 feet per second. a. Assume that the speed of Dr. Lawrence's Impala increases linearly from rest (ie. constant positive acceleration) with a velocity function of v(t) = 11t [in ft/sec]. Write the definite integral for this journey with Dr. Lawrence's Impala. What type of measurement would be produced when evaluating this definite integral? b. Sketch the graph of v(t) on the graph below and shade the area underneath v(t) bounded by the proper interval. Label the axes appropriately. c. Find the area of the shaded region constructed in part b by using a common geometric area formula. d. Find the area of the shaded region constructed in part b by evaluating the definite integral from part a. (Hint: Areas from parts c and d should match) 2. Let's assume now that the acceleration of the car is not a constant positive value, but one that is constantly increasing as well. After all, you don't want to floor the gas pedal from rest, but gradually want to push it down to the floorboard as the car is moving 11 over time. Therefore, the velocity function to consider now is v(t) = =t², which 8. satisfies both the car starting at rest (0,0) and moving at 88 ft/sec after 8 seconds (8,88). a. Write the definite integral for this journey with Dr. Lawrence's Impala. What type of measurement would be produced when evaluating this definite integral? b. Trace the graph of v(t) on the graph below and shade the area underneath v(t) bounded by the proper interval. Label the axes appropriately.