1. **State the problem:** We need to find the average temperature along a 6 meter metal bar where the temperature at a point $x$ meters from one end is given by the function $T(x) = 6x + 12$. 2. **Formula for average value of a function:** The average value $\overline{T}$ of a continuous function $T(x)$ over the interval $[a,b]$ is given by $$\overline{T} = \frac{1}{b - a} \int_a^b T(x) \, dx$$ 3. **Apply the formula:** Here, $a=0$ and $b=6$, so $$\overline{T} = \frac{1}{6 - 0} \int_0^6 (6x + 12) \, dx = \frac{1}{6} \int_0^6 (6x + 12) \, dx$$ 4. **Calculate the integral:** $$\int_0^6 (6x + 12) \, dx = \int_0^6 6x \, dx + \int_0^6 12 \, dx$$ Calculate each separately: $$\int_0^6 6x \, dx = 6 \times \frac{x^2}{2} \Big|_0^6 = 3x^2 \Big|_0^6 = 3 \times 6^2 - 3 \times 0^2 = 3 \times 36 = 108$$ $$\int_0^6 12 \, dx = 12x \Big|_0^6 = 12 \times 6 - 12 \times 0 = 72$$ So, $$\int_0^6 (6x + 12) \, dx = 108 + 72 = 180$$ 5. **Calculate the average temperature:** $$\overline{T} = \frac{1}{6} \times 180 = 30$$ **Final answer:** The average temperature along the bar is $30$ degrees.