1. **State the problem:** Pablo boards a train moving away from the coast at a constant rate. After 11 hours, he is 591 km from the coast, and after 16 hours, he is 836 km from the coast. We need to find the relationship between time and distance and the initial distance from the coast. 2. **Formula and explanation:** Since the train moves at a constant rate, the distance $d$ from the coast is a linear function of time $t$: $$d = mt + b$$ where $m$ is the rate of change (speed) and $b$ is the initial distance from the coast. 3. **Find the rate $m$:** Use the two points $(t_1, d_1) = (11, 591)$ and $(t_2, d_2) = (16, 836)$. $$m = \frac{d_2 - d_1}{t_2 - t_1} = \frac{836 - 591}{16 - 11} = \frac{245}{5} = 49$$ 4. **Write the equation:** $$d = 49t + b$$ 5. **Find the initial distance $b$:** Substitute $t=11$, $d=591$: $$591 = 49 \times 11 + b$$ $$591 = 539 + b$$ $$b = 591 - 539 = 52$$ 6. **Interpretation:** (a) As time increases, Pablo's distance from the coast increases at a rate of 49 kilometers per hour. (b) Pablo was 52 kilometers from the coast when he boarded the train. **Final answers:** - Rate of increase: 49 km/h - Initial distance: 52 km