Please help check my solution to gradient question.

[ozone]

Homework Statement

A wildebeest is charging across a plain. His path takes him to location (x,y) where x is his distance (in miles) east of his starting point and y is his distance in miles north of his starting point at time t. So x and y are functions of t. The air temperature is a function of both location and time.
Right now he is moving at a velocity of 12 miles per hour in the northward direction and 7 miles per hour westward. In the northward direction the temperature changes at a rate of -0.10 degrees per mile, and in the eastward direction the temperature changes at a rate of -0.05 degrees per mile. Also, overall the temperature (irrespective of location) is changing at a rate of -1.6 degrees per hour.
What is the rate of change in air temperature that the wildebeest is experiencing right now?

The Attempt at a Solution

I just want to make sure I set this up right. I first created a function called r(t). We then have

∇r = -7 i + 12 j + 1 k

Next I made a unit vector for the 3 respective changes in temperature
u = -.05/1.603)i - .1/1.603) j + 1.6/1.63 k

Next I just summed the dot product ∇r * u

I just wanted to make sure there are no errors in this solution, and that I set it up in an efficient manner. I have trouble when I have to start mixing (x,y) with t.

Thank you.

 

[HallsofIvy]

I really don't understand at all. if the wildebeast can only move in a two dimensional plane (North-South and East-West), what does "k" represent? And how is the wildebeast moving in that direction at 1 mile per hour?

But then I don't understand the question itself! If the temperature is changing at "-0.10 degrees per mile" in the northern direction and at -0.05 degrees per mile" in the eastern direction, then how can it be changing at -1.6 degrees per hour irrespective of location?

 

[ozone]

Lol that is why I posted it here! A teacher at my college put this on a calc III test, and I want to make sure that I am prepared for an upcoming exam. I chose to set z = 1 in the gradient because I wanted to consider this problem as a 3 dimensional function where the z value would be a shift up and down a 3 dimensional axis perhaps.

However I'm not sure if this would yield a correct answer.

edit : the change in global temperature is what I considered to be our z value.

 

[algebrat]

You could call it r(t)=(x,y,t). Then r'(t)=(-7,12,1)

∇T=(-.05,-.1,-1.6)

Then dT/dt=∇T\cdotr'(t)

Comments:

I'm not sure you need a unit vector.

Rate of change with respect to what, time?

Notice that dT/dt≠∂T/∂t=-1.6