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9.6 - Staying cool / Matching
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<h2>9.6 - Staying Cool</h2>
<p>Let $T(x,y)$ be the temperature in a 10 ft by 10 ft room on a winter night.  One corner of the room is at (0,0) and the opposite corner is at (10,10).  For both of the numbered functions, $T(x,y)$, answer these three questions:
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<li>Draw, or describe in words, or plot with GeoGebra a graph of the temperature function.  Useful prototype commands in GeoGebra: (This should plot one quadrant of a sphere)
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function(sqrt(81-x^(2)-y^(2)), 0, 10, 0, 10)
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z=4
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<p>The last four numbers are the $x$-range to plot, $0\lt x\lt 10$, and then the $y$-range to plot, also 0 to 10. The command <tt>z=4</tt> will display the plane perpendicular to $\uv z$ at a height of $z=4$.  This lets you easily see what part of the surface is above 4 and what part is below 4.

<>Now, for each of the Temperature functions, $T(x,y)$ for this 10 by 10 foot room, plot $z=T(x,y)$.  Use your plot to answer these questions:
<li>Describe the likely floor locations of the heating vents.  Sketch the locations on a 10 by 10 square.
<li>Suppose you like to sleep with a temperature of 70${}^o$ or less.  Where would you put your bed? Again, sketch locations on a square.
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<ol>
<li> $$T(x,y)=78-\frac{1}{10}\left[ x^2+(y-5)^2 \right]$$
<img src="_hw/Cool1.jpg"><br>
Heating vent in middle of left wall.  Sleep below 70 (pink regions)
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<li> $$T(x,y)=\frac12x-y+75$$
<img src="_hw/Cool2.jpg"><br>
Heating vent along bottom, right corner.  Sleep below 70 (pink areas)
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