Problem $4(30$ points$)$:

For a heat treatment process, a $75$ mm $-$thick steel plate $(k=50\frac{W}{m}-K,c_{}=600\frac{J}{k}g-K,$ and $=8000k\frac{g}{m^3})$ needs to be heated from a uniform temperature of $300$ K to a minimum of $1000$ K $.$ It is placed in an oven with a temperature of $1500$ K and a convective heat transfer coefficient of $200\frac{W}{m^2}-K$ on both sides. Determine how long the plate should be left in the oven.