Final answer:
The family must travel an average speed of 67 mph to reach their destination in 5 hours, but since they cannot exceed 65 mph in the second segment, they'll need to drive 7 mph over the 60 mph speed limit during the first segment to make up for time.
Step-by-step explanation:
The family's road trip consists of two segments: the first at a speed limit of 60 mph for 145 miles, and the second at a speed limit of 65 mph for 190 miles. To calculate the speed the family needs to maintain to reach their destination in 5 hours, we will first determine the time it would take to travel each segment at the given speed limits, and then find out how much faster they need to travel to save time.
If they drive at the speed limit:
- First segment: 145 miles ÷ 60 mph = 2.4167 hours
- Second segment: 190 miles ÷ 65 mph = 2.9231 hours
Adding these times gives us: 2.4167 + 2.9231 = 5.3398 hours, which is more than the 5 hours available.
Now, we need to find out how much time to shave off:
5.3398 hours - 5 hours = 0.3398 hours to save
Since the total distance is 145 + 190 = 335 miles, we will need to save time across the total distance:
Total time needed = 5 hours = 300 minutes
Speed required = total distance ÷ total time needed
Speed required = 335 miles ÷ 5 hours = 67 mph
However, since this is an average speed, and they can't go over 65 mph in the second segment, they'll have to go faster in the first segment to make up for it:
Let's assume they go at 67 mph throughout, then reduce speed to 65 mph for the last 190 miles.
Extra speed needed in the first segment solely:
Additional speed = 67 mph - 60 mph = 7 mph over the limit during the first segment
Adjusting for this overspeed in the first segment will affect the time they can spend on the second segment, so a meticulous calculation to figure out the exact speed they need throughout the trip would involve solving a system of equations, which is beyond the scope of this question.