Problem $m_2(17$ points$)$

A graphite cuboid of mass $m=1\mathrm{.}875kg$ is initially at room temperature $25C$ and heated with a heating source $q=1kW$ as shown in Figure $2.$ The dimensions are of the cuboid $10cm15cm5cm$ and the heat transfer coefficient $h=2130\frac{W}{m^2}k.$ Assuming the temperature of the cuboid is going to be homogenous at all times, do the following:

a$)$ Derive the equation of motion for the system.

b$)$ Find the temperature of the cuboid as a function of time.

c$)$ Find the value of $C_p$ if the temperature of cuboid after $10$ seconds is $30C.$

d$)$ Find the time constant of the system and estimate the steady state time.

e$)$ Find the steady state temperature of the cuboid.