1. **Problem statement:** We analyze Kiran's distance from home over time based on the piecewise graph. 2. **Given:** - School is 3 miles from Kiran's home (distance at time 0 is 3). - Noah's house is 1 mile from school towards Kiran's home, so Noah's house is 2 miles from Kiran's home. - Kiran walks from school to Noah's house, stays there, then walks to the library, then home. 3. **a. Total time from school ending to arrival home:** - The graph shows Kiran arrives home near time 3.75 hours (x-axis) when distance is near 0. - So, total time passed is approximately $3.75$ hours. 4. **b. Time stayed at Noah's house:** - Kiran reaches Noah's house at $0.5$ hours (distance 2 miles). - He leaves Noah's house at $2.5$ hours (distance still 2 miles, horizontal line). - Time stayed = $2.5 - 0.5 = 2$ hours. 5. **c. Distance of library from Kiran's house:** - Library is at distance $1.5$ miles from home (y-axis at $3.5$ hours). - So, library is $1.5$ miles from Kiran's house. 6. **d. Kiran's location 3 hours after school ended:** - At $3$ hours, distance from home is about $1.5$ miles (between library and home). 7. **e. Function notation for part d:** - Let $f(t)$ be Kiran's distance from home at time $t$ hours after school ends. - Then $f(3) = 1.5$. 8. **f. When was Kiran walking fastest and how fast?** - Speed is slope of distance vs time. - From school to Noah's house: from $(0,3)$ to $(0.5,2)$, speed = $\frac{3-2}{0.5-0} = \frac{1}{0.5} = 2$ miles/hour. - From Noah's house to library: horizontal line (no movement), speed = 0. - From library to home: from $(3.5,1.5)$ to $(3.75,0)$, speed = $\frac{1.5-0}{3.75-3.5} = \frac{1.5}{0.25} = 6$ miles/hour. - Fastest walking was between library and home at $6$ miles/hour. **Final answers:** - a. $3.75$ hours - b. $2$ hours - c. $1.5$ miles - d. $1.5$ miles from home - e. $f(3) = 1.5$ - f. Fastest walking between library and home at $6$ miles/hour