Course:MATH007/Problem1

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About this problem

This problem was used in a second-year course on multivariable calculus. It tests several things:

  • Understand how a function of two variable can model temperature in a room.
  • Understand what a level curve represents in this context.
  • Know how to sketch level curves.
  • Use the information coming from level curves to answer a question given in the context of the model.

Here this problem makes use of the equation of a circle as all level curves end up being circles. The constants have been chosen to give nice but not too easy answers and provide numbers that make sense in the context of the model.

Keywords

  • multivariable calculus
  • functions of two variables
  • level curves
  • circles
  • temperature
  • word problem (?)


Problem 1

Consider the function

which represents the temperature in a 10 m by 10 m room on a winter night, where one corner of the room is at the point (0,0) and the opposite corner is at (10,10).


  1. Write the equations of the level curves of the temperature function.
  2. Draw the level curves in the square below.
  3. Describe the likely locations of the heat vents.
  4. It is recommended to sleep with a temperature that is between 17°C and 19°C. Shade the region of the room in which you could put your bed.

Solution

1.
Equations of level curves are equations of the form , where is some constant. Therefore the equations of the level curves are So we can see that for different constant temperatures the level curves will just be circles with center and radius . Some nice equations can be derived for . These values of will give circles of radius 1, 7 and 9 respectively.

2.
Rough sketch of level curves

MATH007PROBLEM1.png


3.
As can be seen from the rough sketch, the heat appears to be propgating from the point (0,5), with higher temperatures being closer to that point, therefore we conclude that the heat vents are likely located at the point (0,5).

4.
Refer to sketch to see the proper shaded region.