Question

An Olympic athlete set a world record of 9.14 in the 100-m dash. Did his speed ever exceed 39 km/hr during the race? Explain.

Select the correct choice below and fill in any answer boxes to complete your choice.
(Round to one decimal place as needed.)
A. The average speed is _________km/hr. By the Mean Value Theorem, the speed was exactly _______km/hrat least once. By the Intermediate Value Theorem, all speeds between and______-km/hr were reached, therefore the athlete's speed exceeded 39 km/hr
B. The average speed is __________km/hr. By the Mean Value Theorem, the speed was exactly
at least once. By the Intermediate Value Theorem, all speeds between and ________km/hr were reached, therefore the athlete's speed never exceeded 39 km/hr
C. The average speed is ________km/hr. Since this value is below 39km/hr, it is impossible to tell if his speed exceeded 39 km/hr
.

Answer 1

The sprinter's average speed was approximately 39.4 km/hr, and by the Mean Value Theorem, there must be at least one point where the instant speed matched this average. Therefore, it is likely that the sprinter exceeded 39 km/hr during the race as the top speed must be higher than the average.

Step-by-step explanation:

To determine if an Olympic athlete's speed ever exceeded 39 km/hr during a 100-m dash with a world record time of 9.14 seconds, we need to calculate the average speed and analyze the possibility of the speed exceeding this threshold.

The sprinter's average speed for the race is calculated using the formula speed = distance/time, which equals 100 meters divided by 9.14 seconds or approximately 10.939 m/s. Converting this to kilometers per hour, we multiply by 3.6, giving an average speed of approximately 39.4 km/hr.

According to the Mean Value Theorem, there must be at least one point during the race where the sprinter's instant speed was exactly the average speed. Considering the sprinter starts from rest and accelerates to a top speed before decelerating as they cross the finish line, it is reasonable to infer that his speed did exceed 39 km/hr at some point during the race since the top speed must be greater than the average speed.