1. **State the problem:** We have a piecewise linear model showing a cyclist's distance travelled over time with points A(0,3), B(20,1), and C(50,0). We need to answer several questions about the journey. 2. **How long did it take the cyclist to ride to work?** - The cyclist starts at 0 min and ends at 50 min. - Total time = $50 - 0 = 50$ minutes. 3. **How far did she travel in total?** - Starting distance at A is 3 km, ending at C is 0 km. - Total distance travelled = $3 - 0 = 3$ km (assuming distance decreases as she approaches work). 4. **Find the gradient of [AB] and interpret it:** - Gradient formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ - For points A(0,3) and B(20,1): $$m = \frac{1 - 3}{20 - 0} = \frac{-2}{20} = -0.1$$ - Interpretation: The cyclist is moving towards work at a rate of 0.1 km per minute (distance decreasing). 5. **For how long did the cyclist stop to fix her tyre?** - The graph shows a flat segment between B and C (distance remains constant or changes slowly). - Since B is at 20 min and C at 50 min, but distance decreases, the stop is likely at the flat part between A and B or B and C. - Given the graph points, the cyclist stopped for 30 minutes (from 20 to 50 min) fixing the tyre. 6. **How far had the cyclist travelled after 35 minutes?** - 35 minutes is between B(20,1) and C(50,0). - Gradient between B and C: $$m = \frac{0 - 1}{50 - 20} = \frac{-1}{30} = -\frac{1}{30}$$ - Distance at 35 min: $$y = y_1 + m(x - x_1) = 1 + (-\frac{1}{30})(35 - 20) = 1 - \frac{15}{30} = 1 - 0.5 = 0.5\text{ km}$$ 7. **Write equations of the lines for the 3 parts:** - Segment AB (0 to 20 min): $$y = 3 - 0.1x$$ - Segment BC (20 to 50 min): $$y = 1 - \frac{1}{30}(x - 20) = 1 - \frac{x - 20}{30} = \frac{50 - x}{30}$$ - Since only two segments are given (A to B and B to C), the third segment is not explicitly defined; assuming no third segment. **Final answers:** - a) 50 minutes - c) 3 km - d) Gradient of AB is $-0.1$ km/min, meaning cyclist approaches work at 0.1 km per minute. - e) Cyclist stopped for 30 minutes. - f) Distance after 35 minutes is 0.5 km. - Equations: - AB: $y = 3 - 0.1x$ - BC: $y = \frac{50 - x}{30}$