Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates $(x,y)$ measured in miles. Your global positioning system says your present location is at one of the following points (pick one):

A: $(1,4)$      B: $(4,-9)$      C: $(-4,9)$      D: $(1,-4)$      E: $(2,0)$      F: $(0,3)$

Your guidebook tells you that the height $h$ of the hill in feet above sea level is given by $$h = a - b x^2 - c y^2$$ where $a=5000\hbox{ ft}$, $b=30\,{\hbox{ft}\over\hbox{ mi}^2}$, and $c=10\,{\hbox{ft}\over\hbox{ mi}^2}$.

1. Where is the top of the hill located?
2. How high is the hill?
3. Draw a topographic map of the hill.
   Your map should have at least 3 level curves; label your location on the map.
4. What is your height?
5. Starting at your present location, in what map direction (2-d unit vector) do you need to go in order to climb the hill as steeply as possible?
   Draw this vector on your topographic map.
6. How steep is the hill if you start at your present location and go in this compass direction?
   Draw a picture which shows the slope of the hill at your present location.
7. In what direction in space (3-d vector) would you actually be moving if you started at your present location and walked in the map direction you found above?
   To simplify the computation, your answer does not need to be a unit vector.
8. Stand up and imagine yourself standing at your chosen point on the hill — you will need to decide where the top of the hill is located. Now point in the direction of the gradient at your location.