Answer: i) To find the direction from point P where the temperature increases most rapidly, we need to find the gradient vector of T(x, y) at point P and then determine its direction. The gradient vector of T(x, y) is given by:
∇T(x, y) = ⟨−2x, −4y⟩
Plugging in P(4, -5) into the gradient vector, we get:
∇T(4, -5) = ⟨−8, 20⟩
The direction of the gradient vector is the direction of maximum increase of the temperature at point P. To find this direction, we can normalize the gradient vector by dividing it by its magnitude:
||∇T(4, -5)|| = √((-8)^2 + (20)^2) = 4√29
So the direction of maximum increase of the temperature at point P is:
⟨−8, 20⟩ / (4√29) = ⟨−2/√29, 5/√29⟩
Therefore, the direction from point P where the temperature increases most rapidly is in the direction of the vector ⟨−2/√29, 5/√29⟩.
ii) To find the rate of increase of the temperature at point P, we can take the dot product of the gradient vector at point P with a unit vector in the direction of maximum increase. We already have the normalized direction vector:
⟨−2/√29, 5/√29⟩
Plugging in P(4, -5) into the gradient vector, we get:
∇T(4, -5) = ⟨−8, 20⟩
Taking the dot product of these two vectors, we get:
⟨−8, 20⟩ · ⟨−2/√29, 5/√29⟩ = (-16 + 100)/29 = 84/29
Therefore, the rate of increase of the temperature at point P is 84/29 degrees Celsius per centimeter, rounded to two decimal places.