MHB Find rate of change (Why is my answer wrong?)

[carl123]

The temperature at a point (x, y) on a flat metal plate is given by

T(x, y) = 62/(9 + 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 (1, 1) in the x-direction and the y-direction.

My solution so far:

dT/dx = -124x/(x²+y²+9)²

dT/dy = -124y/(x²+y²+9)²

dT/dx at (1,1) = -1.02 °C/m (x-direction)
dT/dy at (1,1) = -1.02 °C/m (y - direction)

It says my answers is wrong, I don't why? It's also not from a text so i don't know the exact answers

 

[Evgeny.Makarov]

I think your answers are correct up to the second decimal digit. If you are using an automated submission system, it may expect the answer in the form -124/121.

 

[Prove It]

The temperature at a point (x, y) on a flat metal plate is given by

T(x, y) = 62/(9 + 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 (1, 1) in the x-direction and the y-direction.

So wouldn't you need a variable to represent the distance?

$\displaystyle \begin{align*} D^2 = x^2 + y^2 \implies T = \frac{62}{9 + D^2} \end{align*}$

and so

$\displaystyle \begin{align*} \frac{\mathrm{d}T}{\mathrm{d}D} &= -\frac{124D}{\left( 9 + D^2 \right) ^2 } \end{align*}$

Now evaluate this rate at the point (x,y) = (1,1).