JUST SOLVE PROBLEM #3 PLEASE!!!!!

1. (4pt) Back to the cold bald head challenge. If the poor guy is going to be out in the cold for a long time, lets put on a balaclava (a knit hat that covers the head, like in the picture; ignore the eye and mouth holes) to fight the winter cold. The balaclava is 2.3 mm thick, with a thermal conductivity of 0.08 W/mK. The parameters of this problem are a little different than in the first homework, as follows. The ambient air conditions are -9C, with a convective coefficient around the head of 47 W/m²K. Make the following assumptions:

The head is a sphere with diameter of 17.3 cm.

The head is disconnected from the body, so you are working just with the sphere.

There is a constant internal metabolic heat generation of 26,400 W/m³ in the head.

Given all of the above, calculate the steady-state surface temperature of the bald mans scalp. Note: Because the balaclava is very thin, relative to the size of the head, treat the heat transfer through the balaclava as a slab problem. (HINT: Start by calculating the net constant heat flux coming out of the head.)

2. (4pt) Now, let's cover his head with a second/additional balaclava (i.e., a two-layer solution). The second balaclava is 1.6 mm thick with thermal conductivity of 0.11 W/mK. You may again ignore the eye and mouth holes. If you want to keep the head surface temperature (at the interface between the skin and the first layer) to be above 30C at steady-state, what is the lowest ambient temperature for which this double-layer hat solution provides sufficient protection (assuming the same head diameter and convection coefficient as in the previous problem, and still treating the balaclava as slabs),

3. (4pt) Wait... I have another idea. Go back to the problem scenario in HW 4b. Let's create a battery-heated balaclava! So, the single-layer balaclava (2.3 mm thick, k = 0.08 W/mK) from the first problem in HW4b is now electrically-heated, which essentially results in an internal heat generation in the knit material of 84,675 W/m³. Now, with just this new, battery-heated balaclava covering the head (with the same internal and external conditions as in the original problem), what is the steady-state temperature of the scalp?