1. **State the problem:** We are given a graph showing miles traveled over time (hours) for a cyclist. We need to find: a. The interval where the rate of change (speed) is greatest. b. The value of that greatest rate of change. 2. **Recall the formula for rate of change:** The rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the slope formula: $$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$$ Here, $y$ is miles traveled and $x$ is hours. 3. **Analyze the graph segments:** - From 0 to 1 hour: distance goes from 0 to about 15 miles. $$\text{slope} = \frac{15 - 0}{1 - 0} = 15$$ miles per hour. - From 1 to 3 hours: distance goes from 15 to 35 miles. $$\text{slope} = \frac{35 - 15}{3 - 1} = \frac{20}{2} = 10$$ miles per hour. - From 3 to 4 hours: distance stays constant at 35 miles. $$\text{slope} = \frac{35 - 35}{4 - 3} = 0$$ miles per hour. - From 4 to 6 hours: distance goes from 35 to 55 miles. $$\text{slope} = \frac{55 - 35}{6 - 4} = \frac{20}{2} = 10$$ miles per hour. 4. **Determine the greatest rate of change:** The greatest slope is 15 miles per hour, which occurs on the interval from 0 to 1 hour. 5. **Write the answer for part a:** $$0 < x < 1$$ 6. **Write the answer for part b:** The rate of change on this interval is: $$15$$ miles per hour.