1. **State the problem:** We are given two scenarios with scatter plots: Adriana's distance traveled over time and Angelina's calories burned rowing over time. We want to understand the relationships and possibly find formulas for each. 2. **Distance Adriana Traveled:** - Given points near (1,50), (2,90), (3,140). - We want to find the rate of travel and an equation for distance $d$ as a function of time $t$. 3. **Formula and rules:** - Distance traveled at constant speed is $d = rt$ where $r$ is the rate (speed). - To find $r$, use two points: $r = \frac{\Delta d}{\Delta t}$. 4. **Calculate Adriana's rate:** - Using points (1,50) and (2,90): $$r = \frac{90 - 50}{2 - 1} = \frac{40}{1} = 40 \text{ miles per hour}$$ 5. **Check with another point:** - Using (2,90) and (3,140): $$r = \frac{140 - 90}{3 - 2} = \frac{50}{1} = 50 \text{ miles per hour}$$ - Rates differ, so speed is not constant; approximate average rate: $$\text{average } r = \frac{40 + 50}{2} = 45 \text{ miles per hour}$$ 6. **Equation for Adriana's distance:** - Using average rate: $$d = 45t$$ 7. **Calories Burned Rowing:** - Points near (15,50), (30,150), (60,200). - We want calories burned $C$ as a function of time $m$ (minutes). 8. **Formula and rules:** - Calories burned often increase linearly with time: $C = km$ where $k$ is calories per minute. 9. **Calculate rate $k$:** - Using (15,50) and (30,150): $$k = \frac{150 - 50}{30 - 15} = \frac{100}{15} \approx 6.67 \text{ calories per minute}$$ 10. **Check with (30,150) and (60,200):** $$k = \frac{200 - 150}{60 - 30} = \frac{50}{30} \approx 1.67 \text{ calories per minute}$$ - Rates differ, so calories burned rate decreases over time. 11. **Approximate piecewise or average rate:** - Average rate: $$\frac{6.67 + 1.67}{2} = 4.17 \text{ calories per minute}$$ 12. **Equation for calories burned:** - Using average rate: $$C = 4.17 m$$ **Final answers:** - Adriana's distance approximately: $$d = 45t$$ miles, where $t$ is in hours. - Angelina's calories burned approximately: $$C = 4.17 m$$ calories, where $m$ is in minutes.