1. **State the problem:** We need to solve for temperature $T$ in the equation $$30000 = 101325 \left(\frac{T}{288.16}\right)^{-\frac{9.81}{-6.5 \times 10^{-3} \times 287.08}}.$$\n\n2. **Identify the formula and variables:** This equation relates pressure and temperature using the barometric formula. The exponent is $$-\frac{g}{L R}$$ where $g=9.81$, $L=-6.5 \times 10^{-3}$, and $R=287.08$.\n\n3. **Simplify the exponent:** Calculate the exponent value:\n$$-\frac{9.81}{-6.5 \times 10^{-3} \times 287.08} = -\frac{9.81}{-1.86502} = 5.259.$$\n\n4. **Rewrite the equation:**\n$$30000 = 101325 \left(\frac{T}{288.16}\right)^{5.259}.$$\n\n5. **Isolate the power term:** Divide both sides by 101325:\n$$\frac{30000}{101325} = \left(\frac{T}{288.16}\right)^{5.259}.$$\nCalculate the left side:\n$$0.2959 = \left(\frac{T}{288.16}\right)^{5.259}.$$\n\n6. **Solve for $\frac{T}{288.16}$:** Take both sides to the power of $\frac{1}{5.259}$:\n$$\left(0.2959\right)^{\frac{1}{5.259}} = \frac{T}{288.16}.$$\nCalculate the left side:\n$$0.2959^{0.1902} \approx 0.796.$$\n\n7. **Find $T$:** Multiply both sides by 288.16:\n$$T = 0.796 \times 288.16 = 229.3.$$\n\n**Final answer:** $$\boxed{T \approx 229.3}.$$\n\nThis means the temperature $T$ is approximately 229.3 K.