1. **State the problem:** We analyze a piecewise linear distance-time graph of a cyclist's journey to work with points A(0,20), B(20,10), and a third segment continuing to 50 minutes. 2. **How long did it take the cyclist to ride to work?** - The total time is the horizontal axis value at the end of the journey, which is 50 minutes. 3. **How far did she travel in total?** - The total distance is the vertical axis value at the start minus the vertical value at the end. - Start distance at A is 20 km, end distance at 50 min is 0 km (assuming the graph ends at 0 km). - Total distance traveled = 20 km - 0 km = 20 km. 4. **Find the gradient of [AB] and interpret it:** - Gradient formula: $$m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$$ - For segment AB: $$m=\frac{10 - 20}{20 - 0}=\frac{-10}{20}=-0.5$$ - Interpretation: The cyclist is moving at a speed of 0.5 km per minute towards work (distance decreasing). 5. **For how long did the cyclist stop to fix her tyre?** - The handwritten note says "d) 30 min" which likely indicates the stop duration. 6. **How far had the cyclist travelled after 35 minutes?** - After 20 min at B, the cyclist stops or changes speed. - From 20 to 50 min, distance decreases further. - Assuming linear decrease from 10 km at 20 min to 0 km at 50 min: - Gradient for BC: $$m=\frac{0 - 10}{50 - 20}=\frac{-10}{30}=-\frac{1}{3}$$ - Distance at 35 min: $$y=10 + m \times (35 - 20)=10 - \frac{1}{3} \times 15=10 - 5=5\text{ km}$$ 7. **Write equations of the lines for the 3 parts:** - Segment AB (0 to 20 min): $$y=20 - 0.5x$$ - Segment BC (20 to 50 min): $$y=10 - \frac{1}{3}(x - 20) = 10 - \frac{1}{3}x + \frac{20}{3} = \frac{50}{3} - \frac{1}{3}x$$ - Segment before A or after C is not given, so only two segments are defined. **Final answers:** - a) 50 minutes - b) 20 km - c) Gradient of AB is $-0.5$ km/min, meaning the cyclist travels towards work at 0.5 km per minute. - d) 30 minutes stopped - e) 5 km traveled after 35 minutes - Equations: - AB: $y=20 - 0.5x$ - BC: $y=\frac{50}{3} - \frac{1}{3}x$