Question

Suppose that the Celsius temperature at the point (x, y) in the xy plane is T(x,y) = xsin(2y) and that the distance in the xy plane is measured in meters. A particle moving clockwise around the circle of radius 1m centered at the origin at the constant rate of 2 m/s.

A. how fast is the temperature experienced by the particle changing in degrees C per meter at the point P(1/2 , (sqrt[3])/2) ?
B. how fast is the temperature expereinced by the particle changing in degrees C per second at P?

Answer 1

To find the rate at which the temperature experienced by a particle is changing, we calculate the partial derivatives of T(x,y) with respect to x and y and use the chain rule. At the point P(1/2, (sqrt[3])/2), we substitute the given values into the partial derivative expressions.

Step-by-step explanation:

To find the rate at which the temperature experienced by a particle is changing, we need to calculate the partial derivatives of T(x,y) with respect to x and y and then use the chain rule. The partial derivative of T with respect to x is sin(2y) and the partial derivative of T with respect to y is 2x*cos(2y). We can then calculate the total derivative of T with respect to t and divide by the distance traveled in meters to find the rate of change in degrees Celsius per meter. At the point P(1/2, (sqrt[3])/2), we substitute the given values into the partial derivative expressions to find the specific rate of change. To find the rate of change in degrees Celsius per second, we divide the rate of change in degrees Celsius per meter by the speed of the particle..