Question

The temperature at a point (x, y) on a flat metal plate is given by T(x, y) = 14/(5 + x2 + y2), where T is measured in °C and x, y in meters. Find the rate of change of temperature with respect to distance at the point (2, 2) in the x-direction and the y-direction.

Answer 1

Answer with Step-by-step explanation:

We are given that

$T(x,y,z)=\frac{14}{5+x^2+y^2}$

We have to find the rate of change of temperature with respect to distance at the point (2,2) in the x- direction and the y- direction.

$\frac{\partial T'(x,y)}{\partial x}=-\frac{28x}{(5+x^2+y^2)^2}$

$\frac{\partial T'(2,2)}{\partial x}=-\frac{56}{(5+4+4)^2}=-\frac{56}{169}^{\circ} C/m$

$\frac{\partial T'(x,y)}{\partial y}=-\frac{28y}{(5+x^2+y^2)^2}$

$\frac{\partial T'(2,2)}{\partial y}=-\frac{56}{(5+4+4)^2}=-\frac{56}{169}^{\circ} C/m$