1) a heat-seeking particle has the property that at any point (x,y) in the plane

Question

1) a heat-seeking particle has the property that at any point (x,y) in the plane it moves in the direction of maximum temperature increase. If the temperature at (x,y) is T(x,y) = -e-2y cos x, find an equation y = f(x) for the path of a heat-seeking particle at the point (pi/4 , 0)

2) Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases the most rapidly. Based on certain tests, teh concentration of blood (in parts per million) at a point P(x,y) on the surface of seawater is approximated by

C(x,y) = e-(x^2 + 2y^2)/10^4 where x and y are measured in meters in a regular coordinate system with the blood source at the origin.

Suppose a shark is at the point (x0 , y0) when it first detects the presence of blood in the water. Find an equation of the shark's path.

Explanation / Answer

1) The original family of curves is:

e^?2ycosx=C

and the derivative wrt x becomes:

?e^?2ysinx?2e^?2ydydxcosx=0

simplifying we obtain:

dydx=?12tanx

but what we really need is the orthogonal slope which turns into:

dydx=2cotx

solve the family of curves that will give you that derivative. you'll have a constant, but you can set the constant to match your point (pi/4,0). good luck

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