Explanation The system optimal travel time is 30 veh-hr . Given: The total vehicles are 5500 . The speed at route 1 is 65 - mi / h . The speed at route 2 is 50 - mi / h . Formula Used: Write the expression for the first route travel time. t 1 = t 1 ′ + ( 4 x 1 1000 ) …… (I) Here, t 1 is the first route travel time, x 1 is the route flow in thousands of vehicles per hour and t 1 ′ is the free flow travel time for route 1. Write the expression for the second route travel time. t 2 = t 2 ′ + ( x 2 1000 ) 2 …… (II) Here, t 2 is the second route travel time, x 2 is the route flow in thousands of vehicles per hour and t 2 ′ is the free flow travel time for route 2. The expression for free flow travel time of route 1. t 1 ′ = l 1 × 60 v 1 …… (III) Here, l 1 is the length of first route and v 1 is the speed on first route. The expression for free flow travel time of route 2. t 2 ′ = l 2 × 60 v 2 …… (IV) Here, l 2 is the length of second route and v 2 is the speed on second route. Write the expression for user equilibrium travel time. Z = ∫ 0 x 1 t 1 d x + ∫ 0 x 2 t 2 d x …… (V) Here, Z is the user equilibrium time, x 1 is the first route flow rate and x 2 is the second rote flow rate. Write the expression for relation between x 1 and x 2 . x 1 + x 2 = 5500 …… (VI) Write the expression for total route travel time. t = t 1 + t 2 …… (VII) Here, t is total route travel time. Calculation: Substitute 65 - mi / h for v 1 and 7 mi for l 1 in equation (III). t 1 ′ = ( 7 mi ) × 60 65 - mi / h = 420 65 = 6.46 min Substitute 50 - mi / h for v 1 and 4 mi for l 1 in equation (IV). t 2 ′ = ( 4 mi ) × 60 50 - mi / h = 240 50 = 4.8 min Substitute 6.46 min for t 1 ′ in equation (I). t 1 = ( 6.46 min ) + ( 4 x 1 1000 ) Substitute 4.8 min for t 2 ′ in equation (II). t 2 = ( 4.8 min ) + ( x 2 1000 ) 2 Substitute ( 6.46 min ) + ( 4 x 1 1000 ) for t 1 and ( 4.8 min ) + ( x 2 1000 ) 2 for t 2 in equation (V). Z = ∫ 0 x 1 ( ( 6.46 min ) + ( 4 x 1 1000 ) ) d x + ∫ 0 x 2 ( ( 4