Latitude

Task

Standard maps of the earth are broken into a grid of latitude lines (east-west) and longitude lines (north-south). Consider the function, $N(\ell)$, the percentage of earth's surface north of a given latitude, $\ell$ (north of the equator). Several values of $N(\ell)$ (to the nearest tenth) can be determined using the table below.

1. Use the data to sketch a graph of $N(\ell)$ for $30 \leq \ell \leq 90$.
2. Is the graph of $N(\ell)$ increasing or decreasing?
3. What are the units of $\ell$? What are the units of $N(\ell)$?
4. What is the value of $N^{-1}(25)$?
5. Describe what is meant by the expression, $N^{-1}(20)$.

IM Commentary

This task requires students to use data to generate understanding of an invertible function. Some brief notes: First, the table has data ordered by percentage, not latitude, so students will have to reorder the data in order to generate a graph of $N(\ell)$. Second, students are asked to interpret statements about inverse functions, for which an understanding of the quantities' units is particularly helpful. Teacher guidance for interpretation could have students lead their thinking by means of addressing the units involved. Finally, we note that an algebraic description of the function $N(\ell)$ can be expressed succinctly as $N(\ell) = 50(1-\sin(\ell))$. This can be shared with the students at the teacher's discretion.

Finally, we note that due to the geometric significance of this problem, ample opportunities exist for "hook questions" -- for example, it's an interesting question as to which latitude line divides the area of the Northern hemisphere in half. From the table, that answer is precisely the line of latitude at 30 degrees.

Solution

1. 
2. As the latitude increases, the percent area decreases; the function $N(\ell)$ is decreasing.
3. The unit of $\ell$ is degree. The unit of $N$ is percent.
4. Using the chart, when the percent of surface is 25, the latitude is 60 degrees.
5. The expression $N^{-1}(20)$ means the latitude north of which lies 20% of the Earth's surface.