In many countries, automatic number plate recognition is used to catch speeders. The system takes a time-stamped license plate photo at one location (like an on-ramp to a freeway), and then takes a second time-stamped photo at a second location a known distance from the first location (like an exit ramp off a freeway). Since the systems knows both the distance traveled and the time you took to travel that distance, it can determine if, on average, you were speeding, and subsequently mail the car owner a complementary ticket.

Question

In many countries, automatic number plate recognition is used to catch speeders. The system takes a time-stamped license plate photo at one location (like an on-ramp to a freeway), and then takes a second time-stamped photo at a second location a known distance from the first location (like an exit ramp off a freeway). Since the systems knows both the distance traveled and the time you took to travel that distance, it can determine if, on average, you were speeding, and subsequently mail the car owner a complementary ticket.

Let us say you are traveling along such a highway where the speed limit is 105 km/h. Watching the distance markers on the side of the road, you discover at the half-way point you have been traveling at a speed of 117 km/h. What maximum constant speed must you travel for the last half of the trip to avoid your parents (who own the car) getting a speeding ticket in the mail?

Answer 1

To determine the maximum constant speed you must travel for the last half of the trip to avoid getting a speeding ticket, we need to consider the average speed over the entire distance.

Let's assume the total distance of the trip is 'd' kilometers. Since we have been informed that the speed limit is 105 km/h, we can make the following observations:

- For the first half of the trip, you were traveling at 117 km/h.
- Hence, the first half of the trip was completed in (d/2) / 117 hours.
- Similarly, if we assume 'v' as the constant speed for the second half of the trip, the second half of the trip will be completed in (d/2) / v hours.

For the entire trip, the time taken to travel will be the sum of the times taken for the first and second halves. Since the time is the same for both halves (as per the automatic number plate recognition system), we can set up the following equation:

(d/2) / 117 = (d/2) / v

Now, let's solve this equation to find the maximum speed 'v':

(d/2) / 117 = (d/2) / v (Cross-multiplying)

v * (d/2) = 117 * (d/2) (Canceling the common factor of 'd/2')

v = 117 km/h

Therefore, to avoid your parents receiving a speeding ticket in the mail, you must maintain a maximum constant speed of 117 km/h for the last half of the trip.