1. **Problem Statement:** Mukasa travels from town B to village A at 15 km/hr, leaving at 1:06 pm and arriving at 3:06 pm. Okot travels from village A to town B, starting at noon, riding 20 km/hr for 45 minutes, resting 30 minutes, then riding 5 km/hr until arriving at 2:45 pm. 2. **Goal:** Find the distance between town B and village A and analyze the travel times and speeds. 3. **Step 1: Calculate Mukasa's travel time and distance.** Mukasa's travel time = 3:06 pm - 1:06 pm = 2 hours. Speed = 15 km/hr. Distance = speed × time = $15 \times 2 = 30$ km. 4. **Step 2: Analyze Okot's travel segments.** - First segment: 45 minutes = 0.75 hours at 20 km/hr. Distance = $20 \times 0.75 = 15$ km. - Rest: 30 minutes (no distance). - Remaining time: From 12:00 pm to 2:45 pm is 2.75 hours total. Time spent riding after rest = $2.75 - 0.75 - 0.5 = 1.5$ hours. Speed = 5 km/hr. Distance = $5 \times 1.5 = 7.5$ km. 5. **Step 3: Total distance Okot traveled:** $15 + 7.5 = 22.5$ km. 6. **Step 4: Compare distances:** Mukasa's distance = 30 km. Okot's distance = 22.5 km. Since both travel between the same two points, this suggests a discrepancy or different routes. 7. **Step 5: Mathematical graph function:** We can model distance from starting point over time for both travelers. For Mukasa (starting at 1:06 pm, $t$ in hours after 1:06 pm): $$ y = 15t, \quad 0 \leq t \leq 2 $$ For Okot (starting at noon, $t$ in hours after 12:00 pm): $$ y = \begin{cases} 20t, & 0 \leq t \leq 0.75 \\ 15, & 0.75 < t \leq 1.25 \\ 15 + 5(t - 1.25), & 1.25 < t \leq 2.75 \end{cases} $$ This piecewise function models Okot's distance over time. **Final answer:** Distance between town B and village A is 30 km (Mukasa's route).