1. **State the problem:** A motorist travels from town X to town Z with stops and varying speeds, starting at 7:00am. We need to analyze the journey and represent it on a graph. 2. **Analyze the journey segments:** - From X to Y: distance = 40 km, time = 1 hour, speed = 40 km/h. - Rest at Y: 1 hour. - From Y to Z: distance = 50 km, time = 1 hour, speed = 50 km/h. - Rest at Z: 1 hour. - Return from Z to X: speed = 30 km/h, distance = 90 km (40 + 50), time = distance/speed = $$\frac{90}{30} = 3$$ hours. 3. **Calculate timeline:** - Start at 7:00am. - X to Y: 7:00am to 8:00am. - Rest at Y: 8:00am to 9:00am. - Y to Z: 9:00am to 10:00am. - Rest at Z: 10:00am to 11:00am. - Z to X: 11:00am to 2:00pm. 4. **Define the function for distance from X over time:** - From 7:00 to 8:00 (0 to 1 hour): distance increases from 0 to 40 km at 40 km/h. - From 8:00 to 9:00 (1 to 2 hours): distance constant at 40 km (rest). - From 9:00 to 10:00 (2 to 3 hours): distance increases from 40 to 90 km at 50 km/h. - From 10:00 to 11:00 (3 to 4 hours): distance constant at 90 km (rest). - From 11:00 to 14:00 (4 to 7 hours): distance decreases from 90 to 0 km at 30 km/h. 5. **Piecewise function for distance $d(t)$ where $t$ is hours after 7:00am:** $$ d(t) = \begin{cases} 40t & 0 \leq t < 1 \\ 40 & 1 \leq t < 2 \\ 40 + 50(t-2) & 2 \leq t < 3 \\ 90 & 3 \leq t < 4 \\ 90 - 30(t-4) & 4 \leq t \leq 7 \end{cases} $$ 6. **Interpretation:** The graph of $d(t)$ vs $t$ shows the motorist's journey with increasing distance during driving periods and flat lines during rests. Final answer: The motorist's journey is represented by the piecewise function above, starting at 7:00am and ending at 2:00pm after returning to town X.