Peregrine falcons often attack their prey in the air after high-speed dives. Estimates of their top speed in a dive range up to 157 m s^-1 (351 mi hr^-1), although speeds this high have not been accurately measured. Many researchers have investigated the aerodynamic and gravitational forces affecting the flight of these birds. Some have proposed mathematical models to calculate the speed and acceleration of diving falcons.

V.A. Tucker of Duke University has mathematically defined `ideal falcons' and described the equations that model their motion. Suppose the following data were gathered from a diving falcon.

The following graph shows the data from above and the velocity curve for a corresponding `ideal falcon'.

1. Because of air resistance, diving birds are affected by a drag force that limits their top speed. Based on the graph above, estimate the top speed of our falcon. Give your result in miles per hour.
2. Use the trapezoid rule to estimate the distance traveled by our diving falcon over the 20-second time interval.
3. Estimate the average speed of our falcon during its 20-second dive.
4. Use regression to find a low degree polynomial that approximates the velocity function. Sketch the graph of your polynomial function over the interval 0 £ t £ 20.
5. Use your result from Problem 4 to find a function that approximates the altitude loss of our diving falcon. Sketch the graph of your function.
6. Use your result from Problem 5 to estimate the distance traveled by our diving falcon over the 20-second time interval. Compare your result to that of Problem 2.
7. Discuss the advantages and disadvantages of the falcon's high speed diving.