Answer:
80.7 mph
Step-by-step explanation:
In order to catch the car at the exit, the officer must cover the distance in 30 seconds less time than the car does. The time, speed, distance relation can be used.
<h3>Time for car to reach the exit</h3>
The relation between time, speed, and distance is ...
time = distance/speed
Here, speed is in miles per hour, and distance is in miles. That means time will be given in hours by the straightforward application of this formula.
time = (3.6 mi)/(68 mi/h) = (3.6/68) h = 9/170 h
<h3>Time for the motorcycle to reach the exit</h3>
There are 3600 seconds in 1 hour, so 30 seconds represents this fraction of 1 hour:
30/3600 = 1/120
The motorcycle will have this amount of time less than the time the car takes to reach the exit, so ...
(9/170 -1/120) h = (9(120)-170)/(170×120) h = 910/20400 h = 91/2040 h
<h3>Motorcycle speed</h3>
The speed required to cover 3.6 miles in 91/2040 hours is ...
speed = distance/time
(3.6 mi)/(91/2040 h) = 3.6×2040/91 mph = 7344/91 mph ≈ 80.7 mph
The officer's average speed in pursuit must be 80.7 miles per hour (or faster).
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<em>Additional comment</em>
There are other ways the problem can be worked. One of them is to consider the remaining distance the car must travel when the officer begins pursuit. That will be 3.6 mi less (1/120 h)(68 mi/h) = 17/30 miles. The ratio of the officer's speed to the car's speed will be the same as the ratio of the officer's miles to the car's miles: 3.6/(3.6 -17/30) × 68 mph ≈ 80.7 mph.
Intermediate values used in the calculations should be kept at full calculator precision. Rounding should only be done at the end. Here, the actual speed required is 80.7032967... mph, rounded down for the above result.
