The correct answer is: 318.6

Problem Analysis:
Based on the provided figures, we are analyzing a solid slab of thickness $L=0.2\text{m}$ and thermal conductivity $k=10\text{W}/(\text{m}\cdot \text{K})$. Under steady-state conditions, the heat conducted through the slab from Surface 1 (at temperature $T_1$) to Surface 2 (at temperature $T_2=300\text{K}$) must balance the heat lost from Surface 2 to the environment via convection and radiation.
The given data points from the image are:
- Slab thickness, $L=0.2\text{m}$
- Thermal conductivity, $k=10\text{W}/(\text{m}\cdot \text{K})$
- Temperature of Surface 2, $T_2=300\text{K}$
- Convective heat transfer coefficient, $h=100\text{W}/({\text{m}}^2\cdot \text{K})$
- Free stream fluid temperature, $T_{\infty }=293\text{K}$
- Emissivity of Surface 2, $\epsilon =0.5$
- Surroundings temperature, $T_{\text{surr}}=0\text{K}$
- Stefan-Boltzmann constant, $\sigma =5.67\times {10}^{-8}\text{W}/({\text{m}}^2\cdot {\text{K}}^4)$

Energy Balance Equation:
At Surface 2, the energy balance per unit area (heat flux) is given by:
$q_{\text{conduction}}=q_{\text{convection}}+q_{\text{radiation}}$
Substituting the physical laws for each mode of heat transfer:
$\frac{k(T_1-T_2)}{L}=h(T_2-T_{\infty })+\epsilon \sigma ({T_2}^4-{T_{\text{surr}}}^4)$

Step-by-Step Calculation:
1. Calculate the convection heat flux:
$q_{\text{convection}}=100\times (300-293)=100\times 7=700{\text{W/m}}^2$
2. Calculate the radiation heat flux:
$q_{\text{radiation}}=0.5\times 5.67\times {10}^{-8}\times ({300}^4-0^4)$
$q_{\text{radiation}}=0.5\times 5.67\times {10}^{-8}\times (8.1\times {10}^9)=229.635{\text{W/m}}^2$
3. Sum the convective and radiative losses to find the total heat flux leaving Surface 2:
$q_{\text{total}}=700+229.635=929.635{\text{W/m}}^2$
4. Set the conduction heat flux equal to the total heat flux:
$\frac{10\times (T_1-300)}{0.2}=929.635$
$50\times (T_1-300)=929.635$
$T_1-300=\frac{929.635}{50}=18.5927$
$T_1=318.5927\text{K}$

Rounding to one decimal place, we obtain:
$T_1\approx 318.6\text{K}$