Mathematics 241, section 1 – Multivariable Calculus Discussion – Total Accumulation November 10, 2010

Background

Today, we want to begin to consider the problem of computing the “total accumulation” of a function of several variables. We will start by reconsidering the data giving the snow depth at various locations after the “Blizzard of 1978” that we saw before in the context of density plots. Here is the density plot, and the snow depth data that was used to generate it.

Note: In the following, we will consider a simplified version of the actual geometry, in which we ignore the curvature of the Earth. That is, we will treat this region as though it was planar.

Discussion Questions

A) Suppose that the snow in a region (again, planar) of area A square feet is constant of depth = d feet. What is the total volume of snow that fell on that region? (Use units of cubic feet.)

B) If the snow depth is not constant (which of course is always the case), but you have readings of snow depth at several different locations in the region, how could you approximate the total volume of snow that fell on the region? Would the approximation involve any choices on your part? If so, what are they?

C) Based on the snowfall depth data, describe how you could approximate the total accumulation of snow for the New England region in the “Blizzard of 1978.” For the purposes of this, take this region to extend from −74.0 to −70.0 in longitude and 41.0 to 45.0 latitude. (This includes a bit of New York, and excludes most of Maine, but so be it!) Note: In this region, a degree of latitude is about 69 miles and a degree of longitude is about 51 miles. There are 5280 feet in a mile.

D) Carry out your approximation using the numerical data. Note: Not all of the data corresponds to locations in the New England region that you want. The small crosses in the map underlying the density plot indicate the locations of the stations where depth measurements were made.

E) What information would you need to make a better approximation of the total snow accumulation?