CHAPTER 13 PROJECT Measuring Ice Thickness on Crystal Lake

In this project we examine safety issues involving frozen fresh-water lakes. When a lake freezes over, the depth of the ice varies from place to place on the lake. If we can construct a model that gives the depth of ice at any position on a lake, then we can determine whether a given position is safe. Based on studies of ice depth required to support various activities on a lake, Table 1 lists minimum safe ice depths for various loads and activities on clear lake ice.

Our study focuses on Crystal Lake in Otter Tail, MN. Crystal Lake is roughly circular in shape, with a radius of \(0.838 \, {\rm mi}.\)

• 1. Place the center of the lake at the origin \((0,0) \) of a rectangular coordinate system. Using rectangular coordinates \(( x,y) \) measured in miles, write an equation for the shoreline of Crystal Lake. What are the minimum and maximum values of \(x\) and \(y?\) Draw a graph of the lake with the direction north pointing up.

  A simple model for determining the depth \(d\) of ice is \[ d=d( x,y) =\alpha +\alpha \sin ( 3\pi xy) \]

  where \(d\) is measured in inches, \(( x,y) \) are the coordinates of a point on the lake, as given in Problem 1, and \(\alpha\), measured in inches, is a nonnegative constant related to various conditions such as temperature, lake depth, currents, and so on. Suppose on a certain day in December, the value of \(\alpha \) for Crystal Lake is \(\alpha=9\, {\rm in}.\)

• 2. What is the domain of \(d=d( x,y) ?\) What is the range? What are the minimum and maximum values of \(d?\)
• 3. What is the depth of the ice at the center of the lake?
• 4. Graph the level curves of \(d( x,y) =c,\) for \(c=0,3,6,12,15,\) and \(18.\)
• 5.
  1. Using the model, determine whether a snowmobile can safely cross the lake along the path \(y=x.\)
  2. Using the model, determine whether a car can safely cross the lake along the path \(y=-x.\)
• 6.
  1. Find the rate of change of \(d\) at the point \((0.4,-0.1)\) in the direction east (the direction of \(\mathbf {i}\)).
  2. Find the rate of change of \(d\) at the point \((0.4,-0.1)\) in the direction north (the direction of \(\mathbf {j}\)).
  3. Interpret the results obtained in (a) and (b).
• 7.
  1. Find the direction from the point \(\left( 0.4,-0.1\right) \) for which the ice depth increases most rapidly.
  2. Travel a short distance, say \(1{\rm ft},\) in the direction found in (a). At the new point, recalculate the direction for which the ice depth increases most rapidly.
  3. Explain the results obtained in (a) and (b).
• 8.
  1. Find the direction from the point \(( 0.4,-0.1) \) that results in no change in ice depth \(d.\)
  2. Travel a short distance, say \(1{\rm ft},\) in the direction found in (a), and at the new point, recalculate the direction that again results in no change in ice depth.
  3. Explain the results obtained in (a) and (b).
     Another model for the depth of ice on Crystal Lake is given by \[ h=h( x,y) =\alpha \vert x+y\vert +\beta \vert x-y\vert \]
     where \(h\) is measured in inches and \(\alpha \), \(\beta ,\) measured in inches per mile are nonnegative constants, determined by local conditions. Suppose that on a certain day in January, the value of \(\alpha =\beta =9{\rm in}/ {\rm mi}\) for Crystal Lake.

  902

• 9. Sketch the level curves of \(h( x,y) =c,\) for \(c=0,3,6,12\), and \(15.\)
• 10.
  1. Using this model, find the direction from the point \((0.4,-0.1) \) for which the ice depth increases most rapidly.
  2. Travel a short distance, say \(1 {\rm ft},\) in the direction found in (a). At the new point, recalculate the direction for which the ice depth increases most rapidly.
  3. Explain the results obtained in (a) and (b).
• 11.
  1. Using this model find the direction from the point \(\left( 0.4,-0.1\right) \) that results in no change in ice depth \(h.\)
  2. Travel a short distance, say \(1 {\rm ft},\) in the direction found in (a), and at the new point, recalculate the direction for no increase in ice depth.
     A year earlier, the values of \(\alpha \) and \(\beta \) were different. For that day in January, \(\alpha =5\) and \(\beta =12.\)
• 12. Sketch the level curves of \(h( x,y) =5\left\vert x+y\right\vert +10\left\vert x-y\right\vert ,\) for \(c=0,3,6,12,\) and \(15.\)
• 13.
  1. Find the direction from the point \(\left( 0.4,-0.1\right) \) for which the ice depth increases most rapidly.
  2. Travel a short distance, say \(1 {\rm ft},\) in the direction found in (a). At the new point, recalculate the direction for which the ice depth increases most rapidly.
  3. Explain the results obtained in (a) and (b).
• 14.
  1. Find the direction from the point \(( 0.4,-0.1) \) that results in no change in ice depth.
  2. Travel a short distance, say \(1 {\rm ft},\) in the direction found in (a), and at the new point, recalculate the direction for no increase in ice depth.
  3. Explain the results obtained in (a) and (b).
• 15. What conditions might have influenced the difference in \(\alpha \) and \(\beta \) from one year to the next?
• 16. Compare the results obtained from the two models, Which would you choose to model the ice depth of Crystal Lake? Write a brief report that supports your decision.