Q:

The time it takes to reach 63.2% of the maximum temperature inside the a box.

this is for my undergraduate thesis. I think this is a combination of thermodynamics and calculus. This really troubled me for quite a while. So heres the problem.
Given:
 I have an incubator (box) with a volume of 1 m³. The box is made of plywood.
 Initial temperature  = 84° F.
 Power of the heat rod = 200 W.
Question:
What time it takes to reach the air temperature at 63.2% of maximum temperature using the heat rod?
Please comment if I missed out any given or important information to solve this.

A:

The maximum temperature of the box will depend on the rate of heat loss through the box walls.  Otherwise the box will heat up indefinitely.  To know the rate of heat loss (assuming there are no leaks) you need to know the thermal conductivity of the plywood and the surface area of the box (assume a perfect cube).  There isn't one value for "plywood" as there are many different types and thicknesses.  So, this problem as stated, is incomplete.  Btw, the max temperature will be the point at which the heat loss is equal to 200W.

