Applied Homework Problems

 

 

1. A car is traveling at night along a highway shaped like a parabola with its vertex at the origin.  The car starts at a point 1 mile west and 1 mile north of the origin and travels in an easterly direction.  There is a statue located 1 mile east and 0.5 mile north of the origin.  At what point on the highway will the car’s headlights illuminate the statue?  (Hint:  Start by finding the equation of the parabola with a vertex at the origin that passes through the point (-1,1)).

 

 

2. A runner runs around a circular track of radius 100 m at a constant speed of 7 m/s.  The runner’s friend is standing at a distance 200 m from the center of the track.  How fast is the distance between the friends changing when the distance between them is 200 m? (Hint: Law of cosines can help in connecting the rate of change of angle to the rate of change of distance)

 

3. A Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M.  The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M.  Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.  (Hint:  Let f(t) be the onward travel of the monk and let g(t)  be the monk’s trip back to the monastery. Apply intermediate value theorem to the function H(t) = f(t)-g(t).)

 

4.   Two runners start a race at the same time and finish in a tie.  Prove that at some time during the race they have the same velocity.  (Hint:  Using Rolle’s Theorem, consider f(t) = g(t) - h(t) where g and h are the position functions of the two runners.)