Quiz 1 Solutions, Math 223, Section 2, 9-1-98

You are a park ranger in a dangerous mountain area. Late one afternoon, a group of hikers calls in on their cell phone: ``We're stuck up here. One of our friends broke his leg. I'm not quite sure where we are, since we left the trail a long time ago, but if you let the x-axis point to the east and the y-axis point north, then we're at a spot that looks like the graph of z = x² - y² if z is our elevation.'' The cell phone then dies, leaving you out of communication.

1.

On the topographic map of the area, label the location of the hikers with the letter A. The topographic map shows the location of A. Looking at the description of the location of the hikers, we see that the elevation, z, must increase in the x direction and decrease in the y direction. The graph of z = x² - y² is a saddle function. Location A on the map is clearly a saddle between two peaks, one to the east and one to the west.

2.

The ranger station where you are is located at point B. Can you see the location of the hikers from the ranger station using high-powered binoculars? Why or why not? (Ignore the effect of trees.)

While it is true that the ranger station is at a higher elevation than the hikers, someone wasn't thinking clearly about location of the station. Notice that the contour labelled C on the map is higher than the ranger station (7600 feet versus 7500 feet) and thus blocks the view of anything on the other side. Therefore, the ranger cannot see the hikers from the station.

3.

There is a shallow lake with a small island in the center of it. Shade in the lake on the topographic map. Can the hikers see the lake?

The lake is shaded in below. Notice that this lake has a depth between 0 and 100 feet since the entire area lies between two contours at 7000 feet. The island is the tiny area in the middle that is at the height of the shore. Since the hikers at A are at a higher elevation than the lake, and there are no ridges in the way, the hikers can see the lake.

4.

A stream flows from the highest peak on the map into the lake. At any given point along the stream, the water flows into the direction where it is steepest (and always toward lower elevation.) Sketch the path of the stream on the map. Will you have to cross the stream in order to reach the hikers in the shortest amount of time?

The steepest direction is always where the contours are closest together. An approximate path for the stream is drawn below. Since it lies between the station and the hikers, you must cross it in order to approach them.