1. **State the problem:** We are given the temperature function $$T = 5 \sin\left(\frac{\pi}{12} x\right) + 23$$ where $x$ is the number of hours after sunrise. We need to find the rate of change of temperature, i.e., the derivative $$\frac{dT}{dx}$$, at $x=4$ hours. 2. **Formula and rules:** The rate of change of temperature with respect to time is the derivative of $T$ with respect to $x$. Recall the derivative of $\sin(kx)$ is $k \cos(kx)$. 3. **Differentiate the function:** $$\frac{dT}{dx} = 5 \cdot \frac{d}{dx} \sin\left(\frac{\pi}{12} x\right) + \frac{d}{dx} 23 = 5 \cdot \frac{\pi}{12} \cos\left(\frac{\pi}{12} x\right) + 0 = \frac{5\pi}{12} \cos\left(\frac{\pi}{12} x\right)$$ 4. **Evaluate at $x=4$:** $$\frac{dT}{dx}\bigg|_{x=4} = \frac{5\pi}{12} \cos\left(\frac{\pi}{12} \times 4\right) = \frac{5\pi}{12} \cos\left(\frac{\pi}{3}\right)$$ 5. **Calculate $\cos(\pi/3)$:** $$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$ 6. **Substitute back:** $$\frac{dT}{dx}\bigg|_{x=4} = \frac{5\pi}{12} \times \frac{1}{2} = \frac{5\pi}{24}$$ **Final answer:** The rate of change of temperature at $x=4$ is $$\boxed{\frac{5\pi}{24}}$$ which corresponds to option (B).