Final answer:
To cool off the fastest, the bug should move in the direction (-4, -9) and the temperature will drop at a rate of √97 units. To maintain its temperature, the bug should move in the direction (2, 81).
Step-by-step explanation:
To determine the direction in which the bug should move to cool off the fastest, we need to find the direction in which the temperature is decreasing the fastest. We can do this by finding the gradient of the temperature function T(x,y). The gradient is a vector that points in the direction of the steepest ascent of a function. Since we want to find the direction of the steepest descent, we need to find the negative of the gradient.
The gradient of T(x,y) is given by ∇T(x,y) = (-2x, -9y^2). At the point (2,1), the gradient is ∇T(2,1) = (-4, -9). Therefore, the bug should move in the direction (-4, -9) to cool off the fastest.
The rate at which the temperature drops in this direction can be found by taking the dot product of the gradient vector and the unit vector in the direction (-4, -9). The dot product is given by the formula a · b = |a| |b| cos θ, where a and b are vectors and θ is the angle between them.
In this case, |a| = √((-4)^2 + (-9)^2) = √(16 + 81) = √97 and |b| = 1. The angle between a and b is 0 degrees, so cos θ = 1. Therefore, the rate at which the temperature drops in the direction (-4, -9) is |a| |b| cos θ = √97 × 1 × 1 = √97.
(b) To maintain its temperature, the bug should move in the direction opposite to the gradient of T(x,y) at the point (1,3). At this point, the gradient is ∇T(1,3) = (-2, -81). Therefore, the bug should move in the direction (2, 81) to maintain its temperature.