Question: 4. Model with Mathematics. The distance Dan jogged over time is shown in the graph. Pattie’s constant jogging speed is represented in the table. Who has a faster jogging speed? How much more distance will that jogger have traveled with a total of 22 hours jogging? Dan $(10, 90)$ Distance (mi) Time (h) Time (h) Distance (mi) 2 12 3 5 7 42 10 60 A. Use the table to find Pattie’s jogging speed $k$. Complete the table. $k = 60x$ B. Write an equation for the distance in miles $d$ that Pattie jogs in $t$ hours. Graph the equation on the grid with Dan’s graph. $K=60x$ C. Solve the problem and show your work. Graph shape: top-left, a line graph on a coordinate grid with time (h) on the horizontal axis and distance (mi) on the vertical axis, showing an increasing straight line through points including $(10, 90)$.

1. **State the problem:** We are given Dan's jogging data from a graph point $(10, 90)$, meaning Dan jogged 90 miles in 10 hours. Pattie's jogging distances at various times are given in a table, and we need to find her jogging speed $k$, write her distance equation, compare speeds, and find who jogs farther in 22 hours. 2. **Find Pattie's jogging speed $k$ using the table:** From the table, at $t=10$ hours, distance $d=60$ miles. Speed $k$ is distance divided by time: $$k = \frac{d}{t} = \frac{60}{10} = 6 \text{ miles per hour}$$ 3. **Complete the table for Pattie:** Using $k=6$, calculate distances: - At $t=2$: $d=6 \times 2=12$ - At $t=3$: $d=6 \times 3=18$ - At $t=5$: $d=6 \times 5=30$ - At $t=7$: $d=6 \times 7=42$ - At $t=10$: $d=6 \times 10=60$ 4. **Write Pattie's distance equation:** Distance $d$ in miles after $t$ hours is: $$d = 6t$$ 5. **Find Dan's jogging speed:** From Dan's point $(10, 90)$: $$\text{Speed} = \frac{90}{10} = 9 \text{ miles per hour}$$ 6. **Compare speeds:** Dan jogs at 9 mph, Pattie at 6 mph. Dan is faster. 7. **Calculate distances after 22 hours:** - Dan's distance: $$d_D = 9 \times 22 = 198 \text{ miles}$$ - Pattie's distance: $$d_P = 6 \times 22 = 132 \text{ miles}$$ 8. **Find how much more distance the faster jogger traveled:** $$198 - 132 = 66 \text{ miles}$$ **Final answer:** Dan jogs faster and will have traveled 66 miles more than Pattie after 22 hours.