[5 marks] The temperature T(x, y) at points of the xii-plane is given by T(x, y)

Question

[5 marks] The temperature T(x, y) at points of the xii-plane is given by T(x, y) = x^2 - 2y^2. (a) Draw a contour diagram for T showing some isotherms (curves of constant temperature). (b) In what direction should an ant at position (2, -1) move if it wishes to cool off as quickly as possible? (c) If the ant moves in that direction at speed k (units distance per unit time), at what rate does it experience the decrease of temperature? (d) At what rate would the ant experience the decrease of temperature if it moved from (2, -1) at speed k in the direction of the vector -i - 2j? (e) Along what curve through (2, -1) should the ant move in order to continue to experience maximum rate of cooling?

Explanation / Answer

1) x2-2y2 = T would give us the isotherms which would be conventional hyperbolas with a = T and b = 0.5*T

2) The ant should move in the direction opposite to the gradien of the temperature:

i.e. at (2,-1) the gradient would be 2xi - 4yj

or putting values we have, 4i + 4j

   opposite of this is -4i - 4j which means 4*sqrt(2) units decrease in the temperature for a unit distance (1)

Hence the direction of the preferrable movement of the ant would be given by the unit vector of the above,

i.e. -(i + j)/sqrt(2)

3) At the speed k we have k unit distance travelled in unit time,

which means, k*4*sqrt(2) units temperature decrease per unit time ................... By (1)

4) You can do it the same way I did 2nd and 3rd.

The only thing you need to do is to calculate the unit vector along i-2j (which BTW is (i-2j)/sqrt(5)) and take its scalar product with the gradient i.e. (4i+4j) which gives you -4/sqrt(5) and correspondingly the rate of decrease   of temperature would be 4*k/sqrt(5) unit temperature per unit time.

5) Although I am not proving it, the curve it will have to move along, will be a conventional ellipse (with the expression of the form (x2/a2)+(y2/b2)=constant) passing through (2,-1) and having a slope at (2,-1) of 1.

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