Temperature problem in fluids mechanics

[PythagoreLove]

Homework Statement

Assume the temperature of the exhaust in an exhaust pipe can be approximated by
T=T₀(1+ae^-bx)[1+c cos($\omega$t)]
T₀=100^oC,
a=3,
b=0.03m^-1,
c=0.05,
$\omega$=100 rad/s.

If the exhaust speed is a constant 3 m/s, determine the time rate of change of temperature of the fluid particle at x=0 and x=4 m when t=0.

Homework Equations

We know the equation of the temperature in function of the time and the position
T=T₀(1+ae^-bx)[1+c cos($\omega$t)]

The Attempt at a Solution

We know that the time rate of change of temperature of the fluid particle is dT/dt

$\frac{dT}{dt}$=T₀(1+ae^-bx)[-c $\omega$ sin($\omega$t)]

When t=0, dT/dt=0... unfortunately that isn't the answer, since I don't use the exhaust speed (and have no idea how I could use it).

Thank you
PytLove

 

[LawrenceC]

What if you treated x and t as a variables...

 

[PythagoreLove]

Seems like a wonderful idea and totally worked, that problem was so different from the others I've done in fluid mechanics... I have no idea that I needed to use x(t) in my dT/dt.

Thank you Lawrence

PytLove