A particle is located at the point (5, 5) on a metal surface whose temperature at a point (x, y) is T(x, y) = 25 - 3x² - 2y². Find the equation for the trajectory of a particle moving continuously in the direction of maximum temperature increase. y = E) 1

Question

Quizplus · Accepted Answer

To find the direction of maximum temperature increase, we need to find the gradient vector of T(x, y) and normalize the vector. Gradient of T(x, y) is given by: ∇T = [-3², -2²] Then, the magnitude of the gradient vector is: |∇T| = √((-3²)^2 + (-2²)^2) = √(9² + 4²) = √(13²) Therefore, the direction of maximum temperature increase is given by: d = ∇T / |∇T| = [-3², -2²] / √(13²) = [-3²/√(13²), -2²/√(13²)] Therefore, the trajectory of the particle moving in the direction of maximum temperature increase can be written as: x = 5 + t(-3/√13) y = 5 + t(-2/√13) Simplifying this equation, we get: x = (-3/√13)t + 5√13/13 + 5 y = (-2/√13)t + 10√13/13 + 5 Solving for y in terms of x, we get: y = (-2/3)(x - 5) + 25/3 This equation is in the form of y = mx + b, which represents the equation of a straight line. Therefore, the correct choice is B.

A Particle Is Located at the Point (5, 5) on a Metal