Introduction

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Measuring Ice Thickness on Crystal Lake

Minnesota’s nickname is the Land of 10,000 Lakes, and in the winter many of those lakes freeze over. While this might put an end to some water sports like swimming, it opens up a host of other activities for Minnesotans to enjoy–ice skating, ice fishing, ice hockey, ice golf, ice bowling, and ice driving, to name just a few. The annual Art Shanty Projects festival on Medicine Lake even features shacks (complete with radio and heaters), sculptures, printing presses, and concerts, all comfortably resting atop the lake’s frozen surface.

Of course, a frozen lake can only host those people and events that its ice will support. Specific estimates vary, but the recommended ice depth that will safely support one ice skater is about four inches. A dozen or more ice skaters might require closer to 7 inches of ice. A small truck? That would be about 10.5 inches. But how do you determine the depth of the ice on a given lake? Trial and error is prohibitively risky. The Minnesota Department of Natural Resources suggests that you ask at local bait shops or resorts. In the chapter project, we examine another method for measuring ice depth.

CHAPTER 13 PROJECT

In the Chapter Project on page 901, we use two hypothetical models to measure the ice depth in a frozen lake.

In Chapter 12 we defined the partial derivatives \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{\partial y}\) of a function \(z=f( x,y)\) of two variables. One interpretation of \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{\partial y}\) is that they equal the rate of change of \(z\) in the direction of the positive \(x\)-axis and in the direction of the positive \(y\)-axis, respectively. Here, we generalize these ideas to obtain the rate of change of \(z\) in any direction, a directional derivative.

Directional derivatives lead us to the gradient, a vector that is used in finding tangent planes to a surface. The gradient also provides information about paths of quickest ascent (or descent) on a surface.

Finally, we revisit optimization. Just as derivatives are used to find extreme values of functions of a single variable, partial derivatives are used to find extreme values of functions of several variables. We identify techniques for locating extreme values on a surface and investigate applications involving optimization of functions of two variables.