Let's solve the problem step by step. --- ## **Given Data** - **Vehicle speed (\(v\))**: \(15\, \text{m/s}\) - **Cycle length (\(C\))**: \(60\, \text{s}\) - **Effective green time**: half of the cycle length = \(30\, \text{s}\) - **Distances (in meters) between intersections:** - \(D1 = 301\) - \(D2 = 309\) - \(D3 = 304\) - \(D4 = 144\) - \(D5 = 441\) --- ## **Step 1: Calculate Total Distance from Intersection 1 to Intersection 5** \[ \text{Total Distance} = D1 + D2 + D3 + D4 = 301 + 309 + 304 + 144 = 1058\, \text{m} \] --- ## **Step 2: Calculate Travel Time from Intersection 1 to Intersection 5** \[ \text{Travel Time} = \frac{\text{Total Distance}}{\text{Vehicle Speed}} = \frac{1058}{15} \approx 70.53\, \text{s} \] --- ## **Step 3: Determine Ideal Offset** **Definition:** The **offset** between two signals is the time difference between the start of green at the upstream intersection (Intersection 1) and the start of green at the downstream intersection (Intersection 5) that allows a vehicle traveling at the desired speed to reach the downstream intersection just as its signal turns green. \[ \text{Ideal Offset} = \text{Travel Time} \mod \text{Cycle Length} \] \[ \text{Ideal Offset} = 70.53\, \text{s} \mod 60\, \text{s} = 10.53\, \text{s} \] --- ## **Step 4: Final Answer** **The ideal offset of Intersection 5 relative to Intersection 1 is approximately:** \[ \boxed{10.5\, \text{seconds}} \] --- ### **Explanation:** - After a vehicle leaves Intersection 1 at the start of green, it will reach Intersection 5 in 70.53 seconds. - Since the signals repeat every 60 seconds, after 60 seconds the next cycle starts. - Thus, the green at Intersection 5 should start 10.5 seconds after the green at Intersection 1 for the vehicle to hit the green light at both intersections. --- **Summary Table** | Step | Value | |------------------|--------------------| | Total Distance | 1058 m | | Travel Time | 70.53 s | | Cycle Length | 60 s | | Ideal Offset | 10.5 s |