What is the area through which the heat flows at r = 5.92 cm?

[dinospamoni]

Homework Statement

The fluid inside the pipe shown has a temperature of
350 K, but the temperature of the air in the room is only 306 K.
Therefore, heat flows at a constant rate from the fluid, through
the pipe walls, and into the room. The inner pipe radius is 4
cm, and the outer radius is 8 cm. The heat equation is:

dQ/dt = kA(dT/dx)

where x is the direction of heat flow, A is the area through
which the heat flows (i.e., perpendicular to x), and k is the
conductivity of the material through which the heat is flowing.
Determine the temperature of the pipe metal at r = 5.92 cm.

Homework Equations

T(r<4)=350
T(r>8) = 306

The Attempt at a Solution

I've tried this a bunch of times, but can't see to get it. I have done:

Q'=kA(dT/dr)
where Q' is a constant

A=pi*r^2

dT=Q'/(k*pi*r^2) *dr

T=(Q'/k*pi)(-1/r)+c1

I let Q'/k = c2

so

T=c1-c2/r*pi

After imposing the initial conditions:

350 = c1 - c2/4pi

306 = c1 - c2/8pi

from this
c1 = 262
c2= 1105.84

and got the temperature at r=5.92 to be 321.46 K, but this wasn't right.

Any ideas?

I think I went wrong with the initial conditions somewhere

 

[fzero]

A is the area through
which the heat flows (i.e., perpendicular to x),
...

A=pi*r^2

There's a mistake here, since the area through which the heat flows must depend on the length of the pipe.