1. **State the problem:** We are given the distance function $$d(t) = 6t^3 - 12t^2 + 40t$$ where $t$ is time in hours, and we want to find Alan's speed 30 minutes after the start of the race. 2. **Recall the formula:** Speed is the derivative of distance with respect to time, so $$d'(t) = \frac{d}{dt}(6t^3 - 12t^2 + 40t)$$ 3. **Calculate the derivative:** $$d'(t) = 18t^2 - 24t + 40$$ 4. **Convert 30 minutes to hours:** $$30 \text{ minutes} = \frac{30}{60} = 0.5 \text{ hours}$$ 5. **Evaluate the speed at $t=0.5$ hours:** $$d'(0.5) = 18(0.5)^2 - 24(0.5) + 40$$ 6. **Calculate each term:** $$18(0.5)^2 = 18 \times 0.25 = 4.5$$ $$-24(0.5) = -12$$ 7. **Sum the terms:** $$d'(0.5) = 4.5 - 12 + 40 = 32.5$$ 8. **Interpretation:** Alan's speed 30 minutes after the start is $$\boxed{32.5}$$ km/h. Note: The original calculation using $t=30$ is incorrect because $t$ is in hours, not minutes.